Learn about the Subtraction of Rational Numbers with Different Denominators from here. Get to know the detailed procedure to follow while solving Problems on Subtraction of Rational Numbers having Different Denominators. Check out a few examples explaining step by step and get a good hold of the concept and learn to solve problems on your own.

How do you Subtract Rational Numbers with Different Denominator?

Follow the guidelines listed below to Subtract Rational Numbers with Different Denominator. They are in the following fashion

Step 1: Check out whether the Denominator is Positive or not. If either of the Denominators is negative rearrange them so that denominators become positive.

Step 2: Obtain the Denominators of Rational Numbers in the earlier step.

Step 3: Determine the Least Common Multiple of the Denominators of the given Rational Numbers.

Step 4: Express the Rational Numbers in terms of a Common Denominator using the LCM Obtained.

Step 5: Write a Rational Number whose numerator is the difference of numerators of rational numbers obtained in the earlier step and denominator is the LCM obtained in the earlier steps.

Step 6: Rational Number obtained in step 5 is the required rational number. Simplify if it is required.

Solved Examples on Subtracting Rational Numbers with Different Denominator

1. Subtract 9/5 from 5?

Solution:

Given Rational Numbers are 5/1 and 9/5

Since the denominators aren’t the same find the LCM of Denominators.

LCM(1, 5) = 5

Express the Rational Numbers given using the LCM Obtained.

5/1 = 5*5/1*5 = 25/5

9/5 = 9*1/5*1 = 9/5

Subtracting we get 25/5 -9/5

= 16/5

Therefore, 5-9/5 = 16/5

2. Find the Difference of 4/3 and 6/5?

Solution:

Find the LCM of the Denominators 3, 5

LCM(3, 5) = 15

Express the Rational Numbers with Common Denominator using the LCM obtained.

4/3 = 4*5/3*5 = 20/15

6/5 = 6*3/5*3 = 18/15

difference = 20/15 – 18/15

= 2/15

Therefore, Difference of 4/3 and 6/5 is 2/15.

3. Simplify 4/12-5/-6?

Solution:

Since one of the denominators isn’t positive change it to positive by rearranging it.

5/-6 =5*(-1)/-6*(-1)

= -5/6

Find the LCM of the Denominators 12, 6

LCM(12,6) = 12

Express the Rational Numbers in terms of a Common Denominator using the LCM obtained.

4/12 = 4*1/12*1 = 4/12

-5/6 = -5*2/-6*2 = -10/12

Subtracting the Rational Numbers Obtained we get

= 4/12-(-10/12)

= 14/12

= 7/6

The post Subtraction of Rational Numbers with Different Denominator appeared first on Learn CBSE.

If you need assistance with the Subtraction of Rational Numbers then take the help of this article. Check out the steps you need to follow while Subtracting Rational Numbers be them with the Same or Different Denominator. Let us consider a/b, c/d to be two rational numbers then subtracting c/d from a/b is adding the additive inverse of c/d to the rational number a/b.

We can denote the Subtraction of c/d from a/b as a/b – c/d.

Thus we have, a/b – c/d = a/b+(-c/d)[since -c/d is the additive inverse of c/d]

How to Subtract Rational Numbers?

The Procedure to Subtract Rational Numbers having Same and Different Denominators is better explained in the coming modules by considering a few examples. Have a glance at them and solve the Subtraction of Rational Numbers Problems you encounter easily.

Examples

1. Subtract 2/3 from 4/5?

Solution:

Subtracting 4/5 from 2/3

= 4/5-2/3

= 4/5+(-2/3)

= 4*3/5*3+(-2*5/3*5)

= 12/15+(-10/15)

= (12-10)/15

= 2/15

Therefore, 4/5 -2/3 = 2/15.

2. Subtract 3/9 from 2/5?

Solution:

Subtracting 2/5 from 3/9 we get

= 3/9 -2/5

= 3/9+(-2/5)

= 3*5/9*5+(-2*9/5*9)

= 15/45+(-18/45)

= -3/45

= -1/15

Therefore, 2/5-3/9 = -1/15

3. Simplify -6/7-5/8?

Solution:

Given Expression is -6/7-5/8

= -6/7+(-5/8)

= -6*8/7*8+(-5*7/8*7)

= -48/56+(-35/56)

= (-48-35)/56

= -83/56

Therefore, -6/7-5/8 = -83/56

4. What should be subtracted from 4/5 to get the Rational Number 6/15?

Solution:

Let the number to be subtracted as x

From the given data we can write the equation as 4/5-x = 6/15

Rearranging the equation to get the variable x aside we have

4/5-6/15=x

4*3/5*3-6*1/15*1 = x

x= 12/15-6/15

=6/15

The Number to be Subtracted from 4/5 to obtain 6/15 is 6/15.

5. What is the Rational Number to be Added to -7/10 to get 4/7?

Solution:

Let us consider the number to be added as x which results in 4/7

-7/10+x= 4/7

-7/10-4/7 =x

x =-7/10-4/7

= -7*7/10*7-4*10/7*10

= -49/70-40/70

= (-49-40)/70

= -89/70.

Therefore -89/70 when added to -7/10 results in 4/7

6. Sum of Two Rational Numbers is 5/3 if One of the Numbers is 9/20 find the Other?

Solution:

Sum of two rational numbers = One Number + Other Number

5/3 = 9/20+Other Number

Other Number = 5/3-9/20

= 5*20/3*20 – 9*3/20*3

= 100/60-27/60

= 73/60

Therefore, the Other Number to be added to 9/20 to get the sum 5/3 is 73/60.

The post Subtraction of Rational Numbers appeared first on Learn CBSE.

There are a few properties that are applicable while dealing with the Subtraction of Rational Numbers. Check out the Closure, Commutative, Distributive and Associative Properties of Rational Numbers under Subtraction Operation. To help you understand each and every property we have taken enough examples and explained all of them step by step.

Closure Property of Subtraction of Rational Numbers

The Difference between any Two Rational Numbers always results in a Rational Number. Let a/b, c/d be two Rational Numbers then (a/b -c/d) will also result in a Rational Number.

Example

Consider two rational numbers 5/9 and 3/9 then

Subtraction of 5/9-3/9

= 2/9

Therefore, 5/9-3/9 = 2/9 is also a Rational Number.

Commutative Property of Subtraction of Rational Numbers

Subtraction of Two Rational Numbers doesn’t obey Commutative Property. Let us consider a/b, c/d be two rational numbers then (a/b)-(c/d)≠(c/d)-(a/b). Have a look at the Example stated below and verify whether the commutative property is applicable or not.

Example

Consider the rational Numbers 5/8 and 2/8 then

= 5/8 – 2/8

= 3/8

2/8 – 5/8

= -3/8

5/8-2/8≠ 2/8-5/8

Therefore, Commutative Property isn’t applicable for Subtraction.

Associative Property of Subtraction of Rational Numbers

Subtraction of Rational Numbers is not Associative. Let us consider three Rational Numbers a/b, c/d, e/f then (a/b-(c/d -e/f)) ≠ (a/b – c/d) – e/f.

Example

2/8-(4/8-1/8) = 2/8-(3/8)

= -1/8

(2/8-4/8)-1/8 = (-2/8)-1/8

= -3/8

Therefore, 2/8-(4/8-1/8) ≠(2/8-4/8)-1/8.

Distributive Property of Subtraction of Rational Numbers

Multiplication of Rational Numbers is Distributive Over Subtraction. Consider three Rational Numbers then a/b*(c/d-e/f) = a/b*c/d-a/b*e/f.

Example

Consider three Rational Numbers 1/2, 2/3, 4/5 then

1/2(2/3-4/5) = (1/2*2/3 – 1/2*4/5)

= (2/6-4/10)

= (2*5/6*5 – 4*3/10*3)

= (10/30 -12/30)

= -2/30

= 1/2*2/3 – 1/2*4/5

= 2/6-4/10

= -2/30

Therefore, 1/2*(2/3-4/5) = 1/2*2/3 – 1/2*4/5

The post Properties of Subtraction of Rational Numbers appeared first on Learn CBSE.

If you want to Add or Subtract Rational Numbers in an Expression check out the further modules. We have listed the procedure to solve Rational Expressions involving Addition and Subtraction. Check out solved examples of Adding or Subtracting Rational Numbers explaining everything in detail.

How to Solve Rational Expressions Involving Addition and Subtraction?

To Add or Subtract Rational Numbers with the Same Denominator you just need to add/subtract numerators from each other. If the Denominators aren’t the same in the Expressions you need to find a Common Denominator. The Simplest way is to multiply the Denominators with each other. However, this might not have the simplest computations and needs further simplifying afterward.

One way of making computations easier is to find the LCD i.e. common multiple of two or more numbers present.

Solved Examples

1.  Simplify the Rational Expression x/(x+1)+2/(x+1)?

Solution:

Since the Rational Expression has common denominators we can simply add the numerators of each other while keeping the denominators unchanged.

= x/(x+1)+2/(x+1)

= (x+2)/(x+1)

Therefore, x/(x+1)+2/(x+1) on simplifcation will result in (x+2)/(x+1).

2. Simplify Rational Expression 3/(x+1)+2/(x-1)?

Solution:

Since both the denominators aren’t equal we need to find out the LCD. Simply multiply the denominators.

= 3/(x+1)+2/(x-1)

= (3*(x-1)+2(x+1))/(x+1)(x-1)

= ((3x-3)+(2x+2))/(x+1)(x-1)

= (3x-3+2x+2)/(x+1)(x-1)

= (5x-1)/(x+1)(x-1)

Therefore, 3/(x+1)+2/(x-1) on simplifying gives (5x-1)/(x+1)(x-1).

The post Rational Expressions Involving Addition and Subtraction appeared first on Learn CBSE.

In Maths, Rational Numbers sound similar to Fractions and they are expressed in the form of p/q where q is not equal to zero. Any fraction that has non zero denominators is called a Rational Number. Thus, we can say 0 also a rational number as we can express it in the form of 0/1, 0/2 0/3, etc. However, 1/0, 2/0 aren’t rational numbers as they give infinite values.

Continue reading further modules to learn completely about Rational Numbers. Get to know about Types of Rational Numbers, Difference Between Rational and Irrational Numbers, Solved Examples, and learn how to Identify Rational Numbers, etc. In order to represent Rational Numbers on a Number Line firstly change them into decimal values.

Definition of Rational Number

Rational Number in Mathematics is defined as any number that can be represented in the form of p/q where q ≠ 0. On the other hand, we can also say that any fraction fits into the category of Rational Numbers if bot p, q are integers and the denominator is not equal to zero.

• Class 8 Rational Numbers NCERT Solutions
• Rational Numbers Class 8 Extra Questions
• Rational Numbers Class 8 Notes
• NCERT Exemplar Class 8 Maths Rational Numbers
• Class 7 Rational Numbers NCERT Solutions
• Rational Numbers Class 7 Extra Questions
• Rational Numbers Class 7 Notes
• Introduction to Rational Numbers
• What are Rational Numbers?
• Is Every Rational Number a Natural Number?
• Is Zero a Rational Number?
• Is Every Rational Number an Integer?
• Is Every Rational Number a Fraction?
• Positive Rational Number
• Negative Rational Number
• Equivalent Rational Numbers
• Equivalent Form of Rational Numbers
• Rational Numbers in Different Forms
• Properties of Rational Numbers
• Lowest Form of a Rational Number
• Standard Form of a Rational Number
• Equality of Rational Numbers using Standard Form
• Equality of Rational Numbers with Common Denominator
• Equality of Rational Numbers using Cross Multiplication
• Comparison of Rational Numbers
• Rational Numbers in Ascending Order
• Rational Numbers in Descending Order
• Representation of Rational Numbers on the Number Line
• Rational Numbers on the Number Line
• Addition of Rational Numbers with Same Denominator
• Addition of Rational Numbers with Different Denominator
• Addition of Rational Numbers
• Properties of Addition of Rational Numbers
• Subtraction of Rational Numbers with Same Denominator
• Subtraction of Rational Numbers with Different Denominator
• Subtraction of Rational Numbers
• Properties of Subtraction of Rational Numbers
• Rational Expressions Involving Addition and Subtraction
• Simplify Rational Expressions Involving the Sum or Difference

How to Identify Rational Numbers?

You need to check the following conditions to know whether a number is rational or not. They are as follows

• It should be represented in the form of p/q, where q ≠ 0.
• Ratio p/q can be further simplified and expressed in the form of a decimal value.

The set of Rational Numerals include positive, negative numbers, and zero. It can be expressed as a Fraction.

Examples of Rational Numbers

Types of Rational Numbers

You can better understand the concept of sets by having a glance at the below diagram.

• Real numbers (R) include All the rational numbers (Q).
• Real numbers include the Integers (Z).
• Integers involve Natural Numbers(N).
• Every whole number is a rational number as every whole number can be expressed in terms of a fraction.

Standard Form of Rational Numbers

A Rational Number is said to be in its standard form if the common factors between divisor and dividend is only one and therefore the divisor is positive.

For Example, 12/24 is a rational number. It can be simplified further into 1/2. As the Common Factors between divisor and dividend is one the rational number 1/2 is said to be in its standard form.

Positive and Negative Rational Numbers

Arithmetic Operations on Rational Numbers

Let us discuss how to perform basic operations i.e. Arithmetic Operations on Rational Numbers. Consider p/q, s/t as two rational numbers.

Addition: Whenever we add two rational numbers p/q, s/t we need to make the denominator the same. Thus, we get (pt+qs)/qt.

Ex: 1/3+3/4 = (4+3)/12 = 7/12

Subtraction: When it comes to subtraction between rational numbers p/q, s/t we need to make the denominator the same and then subtract.

Ex: 1/2-4/3 = (3-8)/6 = -5/6

Multiplication: While Multiplying Rational Numbers p/q, s/t simply multiply the numerators and the denominators of the rational numbers respectively. On multiplying p/q with s/t then we get (p*s)/(q*t)

Ex: 1/3*4/2=4/6

Division: Division of p/q & s/t is represented as (p/q)÷(s/t) = pt/qs

Ex: 1/4÷4/3 =1*3/4*4 = 3/16

Properties of Rational Numbers

• If we add a zero to a Rational Number you will get the Rational Number Itself.
• Addition, Subtraction, Multiplication of a Rational Number yields in a Rational Number.
• Rational Number remains the same on multiplying or dividing both the numerator and denominator with the same factor.

There are few other properties of rational numbers and they are given as under

• Closure Property
• Commutative Property
• Associative Property
• Distributive Property
• Identity Property
• Inverse Property

Representation of Rational Numbers on a Number Line

Number Line is a straight line diagram on which each and every point corresponds to a real number. As Rational Numbers are Real Numbers they have a specific location on the number line.

Rational Numbers Vs Irrational Numbers

There is a difference between Rational Numbers and Irrational Numbers. Fractions with non zero denominators are called Rational Numbers. All the numbers that are not Rational are Called Irrational Numbers. Rational Numbers can be Positive, Negative, or Zero. To specify a negative Rational Number negative sign is placed in front of the numerator.

When it comes to Irrational Numbers you can’t write them as simple fractions but can represent them with a decimal. You will endless non-repeating digits after the decimal point.

Pi (π) = 3.142857…

√2 = 1.414213…

Solved Examples

Example 1.

Identify whether Mixed Fraction 1 3/4 is a Rational Number or Not?

Solution: The Simplest Form of Mixed Number 1 3/4 is 7/4

Numerator = 7 which is an integer

Denominator = 4 which is an integer and not equal to 0.

Thus, 7/4 is a Rational Number.

Example 2.

Determine whether the given numbers are rational or irrational?

(a) 1.45 (b) 0.001 (c) 0.15 (d) 0.9 (d) √3

Solution:

Given Numbers are in Decimal Format and to find out whether they are rational or not we need to change them into fraction format i.e. p/q. If the denominator is non zero then the number is rational or else irrational.

FAQs on Rational Numbers

1. How to Identify a Rational Number?

If the Number is expressed in the form of p/q where p, q are integers and q is non zero then it called a Rational Number.

2. Is 5 a Rational Number?

Yes, 5 is a Rational Number as it can be expressed in the form of 5/1.

3. What do we get on adding zero to a Rational Number?

On Adding Zero to a Rational Number, you will get the Same Rational Number.

4. What is the difference between Rational and Irrational Numbers?

Rational Numbers are terminating decimals whereas Irrational Numbers are Non-Terminating Decimals.

The post Rational Numbers | Definition, Types, Properties, Standard Form of Rational Numbers appeared first on Learn CBSE.

In order to simplify Rational Expressions involving the Sum or Difference go through the complete article. Have a glance at the step by step procedure for solving Rational Expressions involving the Sum or Difference. The Solved Examples on Rational Numbers will help you get a good grip on the concept. By the end of this article, you can solve problems of Rational Expressions including the Sum or Difference on your own.

How to Simplify Rational Expressions Involving the Sum or Difference?

Follow the below-mentioned guidelines to solve Rational Expressions involving the sum or difference. They are along the lines

Step 1: Firstly, find the LCM of Denominators of all the Numbers Involved.

Step 2: Write the Rational Number whose denominator is the LCM obtained in the earlier step. For obtaining the numerator simply divide the LCM obtained with all the denominators of the rational numbers. Multiply the numerators of the respective rational numbers with the quotients you got. Keep the Addition, Subtraction Signs as it as in the Expressions. Simplify the Expression to get an Integer as Numerator.

Step 3: Reduce the Rational Number to its lowest or simplest form if it is not present. The Rational Number Obtained is the required Rational Number.

Solved Examples Simplifying Rational Expressions involving the Sum or Difference

1. Simplify -3/2 + 9/6 – (-5)/4?

Solution:

Given Expression is -3/2 + 9/6 – (-5)/4

= -3/2+9/6+5/4

Find the LCM of Denominators i.e. 2, 6, 4

LCM(2, 6, 4) = 12

Express the Rational Numbers using the LCM obtained in terms of a common denominator.

-3/2 = -3*6/2*6 = -18/12

9/6 = 9*2/6*2 = 18/12

5/4 = 5*3/4*3 = 15/12

Placing the Rational Numbers in the Expression we get

= -18/12+18/12+15/12

= 15/12

Therefore, -3/2 + 9/6 – (-5)/4 on simplifying gives a Rational Number 15/12.

2. Simplify 7/10 – (-5)/14 + 9/-3?

Solution:

Given Rational Expression is 7/10 – (-5)/14 + 9/-3

As One of the Denominators is negative we rearrange it to get a positive denominator.

9/-3 = 9*(-1)/-3*(-1) = -9/3

The Rational Expression becomes 7/10+5/14-9/3

Find the LCM of Denominators 10, 14, 3

LCM(10, 14,3) = 210

Express the Rational Numbers in terms of a Common Denominator using the LCM obtained.

7/10 = 7*21/10*21 = 147/210

5/14 = 5*15/14*15 = 75/210

-9/3 = -9*70/3*70 = -630/210

Placing the Rational Numbers in the Expression we get

= 147/210+75/210-630/210

= (147+75-630)/210

= -408/210

= -68/35

Therefore, on simplifying 7/10 – (-5)/14 + 9/-3 we get -68/35.

The post Simplify Rational Expressions Involving the Sum or Difference appeared first on Learn CBSE.

If you are looking for help on Properties of Multiplication of Rational Numbers you have come the right way. Check out Closure, Commutative, Associative, Existence of Multiplicative Inverse Property, Distributive Property, Multiplicative Property of 0. Get to Know the Multiplication of Rational Numbers Properties along with few examples and get a better idea of the concept.

Closure Property of Multiplication of Rational Numbers

Rational Numbers are closed under Multiplication. Let us assume two rational numbers a/b, c/d then their product (a/b*c/d) is also a Rational Number. Check out a few examples listed to understand the Closure Property better.

Examples

(i) Consider the two rational numbers 1/3 and 4/7 then

1/3*4/7

= 4/21

Therefore, 4/21 is also a Rational Number.

(ii) Consider two rational numbers -3/4 and 5/8

= -3/4*5/8

= -15/32

Thus, Product or Multiplication of Rational Numbers -3/4, 5/8 is also a Rational Number -15/32.

Commutative Property of Multiplication of Rational Numbers

Multiplication of Rational Numbers is Commutative. Two Rational Numbers can be multiplied in any order. Let us consider two rational numbers a/b, c/d then

(a/b*c/d) = (c/d*a/b)

Example

(i) Consider Two Rational Numbers 3/4 and 5/2 then

(3/4*5/2) = 3/4*5/2 = 15/8

(5/2*3/4) = 5/2*3/4 = 15/8

Therefore, (3/4*5/2) = (5/2*3/4)

(ii) Consider Two Rational Numbers -4/5 and -3/7

(-4/5*-3/7) = (-4*-3)/35 = 12/35

(-3/7*-4/5) =(-3*-4)/7*5 = 12/35

Therefore, (-4/5*-3/7) = (-3/7*-4/5)

Associative Property of Multiplication of Rational Numbers

Rational Numbers obey the Associative Property of Multiplication. While Multiplying Three or More Rational Numbers they can be grouped in any order. Consider Rational Numbers a/b, c/d, e/f then we have (a/b × c/d) × e/f = a/b × (c/d × e/f).

Example

Consider Rational Numbers 1/2, 4/5, 6/4 then

(1/2*4/5)*6/4 =(1*4/2*5)*6/4

= (4/10)*6/4

= 4*6/10*4

= 24/40

1/2*(4/5*6/4) = 1/2*(4*6/5*4)

= 1/2*(24/20)

=24/40

Therefore, (1/2*4/5)*6/4 = 1/2*(4/5*6/4)

Existence of Multiplicative Identity Property

For a Rational Number a/b we get (a/b*1) = (1*a/b) = a/b. 1 is called the Multiplicative Identity of Rationals.

Example

(i) Consider the Rational Number 4/3 we have

(4/3*1) = (4/3*1/1) = (4*1/3*1) = 4/3

(1*4/3) = (1/1 *4/3) = (1*4/1*3) = 4/3

(4/3*1) = (1*4/3) = 4/3

Existence of Multiplicative Inverse Property

Nonzero rational number a/b has multiplicative inverse b/a.

(a/b*b/a) = (b/a*a/b) = 1

b/a is called the Reciprocal of a/b. Zero has no reciprocal.

Example

(i) Reciprocal of 4/7 is 7/4 since 4/7*7/4 = 1

(ii) Reciprocal of -5/3 is -3/5 since -5/3*-3/5 = 1

Distributive Property of Multiplication over Addition

Let us assume three rational numbers a/b, c/d, e/f . Distributive Property states that a/b*(c/d+e/f) = (a/b*c/d+a/b*e/f)

Example

Consider three Rational Numbers 1/3, 4/5, 7/8 then

1/3*(4/5+7/8) =1/3*((4*8+7*5)/40)

= 1/3*(67/40)

= 67/120

(1/3*4/5+1/3*7/8) = (1*4/3*5+1*7/3*8)

= (4/15+7/24)

= (4*8+7*5)/120

= 32+35/120

= 67/120

Therefore, 1/3*(4/5+7/8) = (1/3*4/5+1/3*7/8)

Multiplicative Property of 0

Any Rational Number multiplied with 0 gives 0. For a Rational Number a/b, we have(a/b*0) = (0*a/b) = 0

Example

(i) (4/7*0) = (4/7*0/1) = (4*0/7*1) = 0

Similarly (0*4/7) = (0/1*4/7) =0

(ii)(-12/5*0) =(-12/5*0/1) =(-12*0/5*1) = 0

Similarly (0*-12/5) = (0/1*-12/5) = 0*-12/1*5 = 0

The post Properties of Multiplication of Rational Numbers appeared first on Learn CBSE.

If you ever need assistance with Rational Expressions including Addition, Subtraction, Multiplication Operations you have come the right way. We have covered the step by step process of simplifying Rational Numbers involving Addition, Subtraction, Multiplication. Check out the solved examples and get a grip on the concept.

Solved Examples on Rational Expressions Involving Addition, Subtraction and Multiplication

1. Simplify the Rational Expression (-4/7*3/2)-(9/4*5/3)

Solution:

Given Rational Expression is (-4/7*3/2)-(9/4*5/3)

=  (-4*3/7*2)-(9*5/4*3)

= (-12/14)-(45/12)

= -12/14-45/12

= (-12*6-45*7)/84

= (-72-315)/84

= -387/84

2. Simplify the Rational Expression (-2/3 × 5/4) + (9/4 × 10/2) – (1 × 3/4× 4)

Solution:

Given Rational Expression = (-2/3 × 5/4) + (9/4 × 10/2) – (1 × 3/4× 4)

= (-2*5/3*4)+(9*10/4*2)-(12/4)

= (-10/12)+(90/8)-12/4

= -10/12+90/8-12/4

= (-10*2+90*3-12*6)/24

= (-20+270-72)/24

= 178/24

= 89/12

3. Simply the Rational Expression (-7/12 × 11/(-7)) – (1 × (1/3)) + (1/4 × 1/4)

Solution:

Given Rational Expression = (-7/12 × 11/(-7)) – (1 × (1/3)) + (1/4 × 1/4)

= (-7/12 × 11/(-7)) – (1/3) + (1/16)

= (-7*11/12*-7)-(1/3) + (1/16)

= -77/-84-1/3+1/16

= 77/84-1/3+1/16

=(77*4-1*112+1*21)/336

=(308-112+21)/336

= 217/336

= 31/48

The post Rational Expressions Involving Addition, Subtraction and Multiplication appeared first on Learn CBSE.

In general, the Rational Number Obtained after interchanging the Numerator and Denominator is called Reciprocal of a Rational Number. Go through the entire article to learn how to find the Reciprocal of a Rational Number. Check out examples of finding the Reciprocal of Rational Numbers in the further sections and solve related problems easily.

How to find Reciprocal of a Rational Number?

Reciprocal of a Rational Number means interchanging of numerator and denominator. Let us assume a Non-Zero Rational Number a/b there exists a rational number b/a such that

a/b*b/a=1

Here the rational number b/a is called the Reciprocal or Multiplicative Inverse of a/b and is denoted by (a/b)^-1.

Solved Examples on finding Reciprocal or Multiplicative Inverse

1. Find the Reciprocal of -3/2?

Solution:

Given Rational Number is -3/2

Numerator = -3

Denominator = 2

Interchanging the Numerator and Denominator to obtain the Reciprocal i.e. we have the Numerator = 2, Denominator = -3

Resultant Rational Number = 2/-3

Reciprocal of a Rational Number is 2/-3.

2. Find the reciprocal of 3/11*4/5?

Solution:

Given Expression is 3/11*4/5

= 3*4/11*5

= 12/55

Reciprocal of 12/55 is 55/12 i.e. after Interchanging the Numerator and Denominator.

3. Find the reciprocal of -4/3 × 7/-8?

Solution:

Given Rational Expression is -4/3 × 7/-8

= -4*7/3*-8

= -28/-24

= 7/6

The reciprocal of 7/6 is 6/7.

The post Reciprocal of a Rational Number appeared first on Learn CBSE.

The mathematical Operation of dividing one rational number with other rational number is called the Division of Rational Numbers. Check out the detailed procedure for solving problems on Dividing Rational Numbers. See Examples related to the Rational Numbers Division and learn how to solve various questions related easily. Usually, a Rational Number can’t be divided by another one due to its complexity.

How to Divide Rational Numbers?

Follow the below guidelines to solve problems on the Division of Rational Numbers easily. They are along the lines

Step 1: Firstly, express the given Rational Numbers in the form of a fraction.

Step 2: Keep the numerator part as it is and multiply with the reciprocal of the denominator in rational number.

Step 3: Find the Product of the Rational Numbers which is nothing but the Division of Rational Numbers.

Let us consider m, n to be two rational numbers then m ÷ n = m*1/n. The Dividend is the number to be divided i.e. m whereas Divisor is the number dividing the dividend i.e. n. When Dividend is divided by the Divisor the result is called Quotient. Do remember Division by 0 is not defined when you are solving related problems.

Solved Examples on Rational Numbers Division

1. Divide Rational Numbers 9/12 and 5/4?

Solution:

Given Rational Numbers are 9/12 and 5/4

= 9/12÷5/4

= 9/12*4/5

= 9*4/12*5

= 36/60

= 3/5

2. Divide Rational Numbers -3/25 by 4/5?

Solution:

Given Rational Numbers are -3/25 and 4/5

= -3/25÷4/5

= -3/25*5/4

=-3*5/25*4

= -15/100

= -3/20

3. Simplify -7/40 ÷ (-2)/8?

Solution:

= -7/40 ÷ (-2)/8

= -7/40*8/-2

= -7*8/40*-2

= -56/-80

= 7/10

4. Simplify 10/22 ÷ (-5)/8?

Solution:

= 10/22 ÷ (-5)/8

= 10/22*8/-5

= 10*8/22*-5

= 80/-110

= 8/-11

The post Division of Rational Numbers appeared first on Learn CBSE.

Rational Expressions Division is the same as the division of Fractions. While dividing a Fraction you will just flip and multiply. Simply change the dividing by a fraction to multiplying by that fraction’s reciprocal. Get to know the Rational Expressions Involving Division Step by Step Procedure. We even jotted a few examples on Rational Numbers Division explaining everything in detail.

Procedure for Dividing Rational Expressions

The Rational Expressions Division is similar to Fractions Division. The step by step procedure is listed as under

• You just need to flip and multiply just like Fractions during the division operation.
• Later to simplify the multiplication factor the numerators and denominators.
• Later check for any duplicate factors and cancel out them.

Solved Examples on Dividing Rational Expressions

1. Divide 3/7 by 9/45?

Solution: 

= 3/7 ÷ 9/45

= 3/7*45/9

= 3*45/7*9

= 135/63

2. Simplify 3x⁴/4÷9x/2?

Solution: 

= 3x⁴/4÷9x/2

= (3x⁴/4)*(2/9x)

= (3.x.x³/4)*(2/9x)

Canceling out the common term x we get the Rational Expression as follows

= (3x³/4)*(2/9)

= (3x³*2)/4*9
= 6x³/36
= x³/6

3. Divide and Simplify the Result (x+4)/(x²-16)÷(x-1)/(x²-4x+3)?

Solution:

Given Rational Expression = (x+4)/(x²-16)÷(x-1)/(x²-4x+3)

((x+4)/(x²-16))*((x²-4x+3)/(x-1))

Factoring out the numerators and denominators we have

= (x+4)/(x+4)(x-4)*(x-3)(x-1)/(x-1)

Canceling out the duplicate factors we get

= 1/(x-4)*(x-3)/1

= 1*(x-3)/(x-4)*1

= (x-3)/(x-4)

The post Rational Expressions Involving Division appeared first on Learn CBSE.

If you ever need assistance solving the Rational Expressions Involving Division you can have a look at the Properties of Division of Rational Numbers prevailing. Make use of the Rational Numbers Division Properties to simplify the expressions during your calculations. Each of the Properties is explained in detail taking enough examples. To gain adequate knowledge and solve related problems on your own.

Closure Property of Division of Rational Numbers

Rational Numbers are closed under Division Except for Zero. Let us assume two rational numbers a/b, c/d where c/d ≠0 then (a/b ÷ c/d) is also a Rational Number.

Example

(i) 2/3÷ 4/5 = 2/3*5/4

= (2*5)/3*4

= 10/12

Therefore 2/3÷ 4/5 i.e. 10/12 is also a Rational Number.

(ii) 3/7 ÷ -5/4

Clearly -5/4 ≠0

= 3/7*-4/5

= 3*-4/7*5

= -12/35

Therefore, 3/7 ÷ -5/4 i.e. -12/35 is also a Rational Number.

Closure Property is true for division except for zero.

Commutative Property of Division of Rational Numbers

Division of Rational Numbers isn’t commutative. Consider two rational number a/b, c/d then a/b÷c/d ≠ c/d÷a/b

Example

1/4÷3/2 = 1/4*2/3 = 1*2/4*3 = 2/12 = 1/6

3/2÷1/4 = 3/2*4/1 = 3*4/2*1 = 12/2 = 6

Therefore, 1/4÷3/2 ≠3/2÷1/4

Thus, Commutative Property is not true for Division.

Associative Property of Division of Rational Numbers

Usually, the Division of Rational Numbers doesn’t obey the Associative Property. Let us consider a/b, c/d, e/f be three Rational Numbers then a/b ÷ (c/d ÷ e/f) ≠ (a/b ÷ c/d) ÷ e/f

2/5÷(4/5 ÷ 1/9) = 2/5÷(4/5*9/1) = 2/5÷(36/5) = 2/5*5/36 = 2*5/5*36 = 10/180 = 1/18

(2/5÷4/5) ÷ 1/9 = (2/5*5/4)÷ 1/9 = (2*5/5*4) ÷ 1/9 = 10/20÷ 1/9 = 10/20*9/1 = 10*9/20*1 = 90/20 = 9/2

2/5÷(4/5 ÷ 1/9) ≠ (2/5÷4/5) ÷ 1/9

Property of 1 of Division of Rational Numbers

For every rational number a/b we have (a/b÷1) = a/b

Example

(i) 12/5÷1 = 12/5

(ii) 4/-7÷1= 4/-7

(iii) 3/9÷1= 3/9

Property 1

For any non zero rational number a/b we have (a/b÷a/b) =1

Example

(i) 3/4÷3/4 = 3/4*4/3 = 3*4/4*3 = 12/12 =1

(ii) 4/7÷4/7 = 4/7*7/4 = 4*7/7*4 = 28/28 = 1

The post Properties of Division of Rational Numbers appeared first on Learn CBSE.

A Rational Number is a number that can be expressed in the form of p/q where q ≠ 0. If you want help in inserting  Rational Numbers between Two Rational Numbers make use of this article. Check out the Step by Step Process to find the Rational Numbers between Two Rational Numbers of either Same or Different Denominator. Have a glance at the solved examples and understand the concept better.

We can insert infinitely many rational numbers between Two Rational Numbers. This Property of Rational Numbers is called Dense Property.

How to find Rational Numbers between Two Rational Numbers with the Same Denominator?

• The first and foremost step while finding the Rational Numbers between Two Rational Numbers is to check the values of Denominators.
• In the case of the same denominators check the value of the numerator.
• If the numerators differ by a large value then you can simply write the rational numbers with an increment of one while keeping the denominator part unaltered.
• If the numerators differ by a smaller value than the number of rational numbers to be found simply multiply the numerators and denominators by multiples of 10.

Examples

Find 4 Rational Numbers between 1/8 and 6/8?

Solution:

Since the Denominators of Two Rational Numbers are Same and Numerators differ by a large value increment the numerator by 1 while keeping the Denominator Unaltered.

2/8, 3/8, 4/8, 5/8 are the 4 Rational Numbers that lie between 1/8 and 6/8.

Find 10 Rational Numbers are to be found between 1/5, 4/5?

Solution:

If 10 Rational Numbers are to be found between 1/5, 4/5 both the rational numbers are to be multiplied by 10.

1/5*10/10 = 10/50

4/5*10/10 = 40/50

10 Rational Numbers between 10/50 and 40/50 are 11/50, 12/50, 13/50, 14/50, 15/50, 16/50, 17/50, 18/50, 19/50, 20/50.

How to find Rational Numbers between Two Rational Numbers with the Different Denominators?

• To find Rational Numbers between Two Rational Numbers with the Different Denominators you need to equate the Denominators firstly.
• You can Equate the Denominators by finding their LCM or by multiplying the denominators of one to another one’s numerator and denominator.

Example

1. Find Rational Numbers between 1/2 and 3/4?(At least 5)

Solution:

Since the Rational Numbers given doesn’t have the same denominator firstly equate the denominators.

Equating the Rational Numbers we have

1/2*4/4 = 4/8

3/4*2/2 = 6/8

We can’t write 5 Rational Numbers between 6/8 and 4/8. Multiply the Numerators and Denominators by multiples of 10.

4/8*10/10 = 40/80

6/8*10/10 = 60/80

5 Rational Numbers between 40/80 and 60/80 are 41/80, 42/80, 43/80, 44/80, 45/80.

The post Rational Numbers between Two Rational Numbers appeared first on Learn CBSE.

A Rational Number is a Number that can be represented in the form of p/q where the denominator q is not equal to zero. In this article of ours, you will learn how to find Rational Numbers between Two Rational Numbers. Check out the entire procedure of finding Rational Numbers between Two Rational Numbers as well as the Solved Examples. After going through the article, we are sure you can solve the related problems on your own.

Procedure for finding Rational Numbers between Two Rational Numbers

Let us consider x, y to be Two Rational Numbers. One of the simplest methods of finding a Rational Number Between Two Rational Numbers is to find the average of the given Rational Numbers i.e.  (x+y)/2. One can find as many rational numbers as they want between two given rational numbers by calculating the average between them.

For better understanding, we have provided a few examples. Have a glance at them and understand the process even better.

Solved Examples

1. Find out a Rational Number between 2/5 and 3/2?

Solution:

Given Rational Numbers are 2/5 and 3/2

Find the Average of the given rational numbers to determine the rational number between them

= (2/5+3/2)/2

= (4/10+15/10)/2

= (19/10)/2

= 19/10*1/2

= 19/20

Therefore, 19/20 is a Rational Number between 2/5 and 3/2.

2. Find out the Rational Number lying between 1/4 and 1/2?

Solution:

Find the average of given rational numbers to obtain the Rational Number between them

= 1/2(1/4+1/2)

= 1/2((1+2)/4)

= 1/2*(3/4)

= 3/8

3/8 is a Rational Number between 1/4 and 1/2.

3. Find three Rational Numbers lying between 4 and 5?

Solution:

Find the average of given rational numbers to obtain the Rational Number between them

= 1/2(4+5)= 9/2

Then 4< 9/2 < 5

Rational Number between 4 and 9/2 is

= 1/2(4+9/2)

= 1/2((8+9)/2)

= 1/2(17/2)

= 17/4

Rational Number between 9/2 and 5 is

= 1/2(9/2+5)

= 1/2((9+10)/2)

= 1/2(19/2)

= 19/4

3< 17/4 < 9/2 < 19/4

Therefore, three Rational Numbers between 4 and 5 are 9/2, 17/4, 19/4

The post How to Find Rational Numbers? appeared first on Learn CBSE.

Class 12th Organic Chemistry Syllabus, Preparation Plan is here. Clear all your doubts about How to Study Organic Chemistry Class 12? We are providing a detailed plan and guide to score 90+ marks in Organic Chemistry Class XII. Refer to Class 12 Chemistry Study Material for better preparation. Go through the below sections to find many important points that help you to make your preparation better.

How to Study Organic Chemistry Class 12?

Do you feel Organic Chemistry tough? No worries! We are here to guide you from beginning to end in your Organic Chemistry preparation. Moreover, we will give you tips to memorise chemical formulae and reactions. Most of the candidates give a lot of importance to Chemistry while preparing for Class 12th exams. So, they feel panic and stressed about how to score top marks in this subject. Actually, preparing for Organic Chemistry is very easy. You just have to concentrate and be thorough with all the topics and concepts.

Do not go with bookish knowledge, rather you understand the concepts and start your preparation from the beginning. Every character in Organic Chemistry has some hidden story. You just have to follow the logic and need good memorization to excel at it. You have to apply as much as you remember. Take time and prepare each chapter thoroughly. Check the best preparation plan to follow all the steps and reach your target.

Books for Class 12th Organic Chemistry

• Concepts of Organic Chemistry by Tej Prakash Akhouri
• Concepts of Organic Chemistry for Competitive Examinations Vol. I 2020-21 by Bharti Gupta, Ajnish Kumar Gupta
• Handbook of Chemistry by Singh Rp
• Together with Chemistry Study Material for Class 12 by R. P Manchanda, Shivansu Manchanda
• Organic Chemistry by Robert Thornton Morrison, Robert Neilson Boyd, Saibal Kanti Bhattacharjee
• Modern Approach To Chemical Calculations An Introduction To The Mole Concept by Ramendra C. Mukerjee
• Organic Reactions Conversions Mechanisms & Problems by R L Madan

12th Class Chemistry Syllabus

• Alcohols, Phenols and Ethers
• Polymers
• Chemistry in Everyday Life
• Haloalkanes and Haloarenes
• Biomolecules
• Amines
• Aldehydes, Ketones and Carboxylic Acids

Class 12 Organic Chemistry Preparation Tips

Here we are giving you the tips on How to Study Organic Chemistry Class 12? Check below

• If you are not really good at Chemistry, then you have to spend a lot of time learning and practicing each topic separately.
• Collect the complete syllabus and books to start your preparation.
• Do not rush to complete your syllabus fast, indeed start preparing on beforehand. You will require a lot of time to memorize formulae, scientific concepts and reactions.
• Follow many textbooks to understand the topic more clearly. It is true that you will remember what you see more than what you hear. So, try visualizing all the concepts as you learn them.
• If you get any doubts or need any clarifications, get them as soon as possible from youtube videos, lecturers, peers etc.
• Start attending laboratories and classes to memorize them practically. You can easily score marks in labs as they are easy to perform and understanding those topics will help you in answering theoretical questions.
• Prepare a timetable and make sure you allocate time to prepare all concepts and chapters.
• Grasp all the chemical formulae and equations.
• Also, attend several mock tests and answer yourself whether you are on the right track or not. Know all your strengths and weaknesses before going to the exam.
• Don’t ignore numerical problems. You may find some of the direct questions or some questions that are mixed up with the concepts. If they are direct questions, you can easily score marks.
• Revise at least 2-3 times after completing the preparation.
• Solve chemistry previous question papers and create an exam environment around so that you can improve your time management skills.
• Know the marking scheme and duration of time beforehand to avoid time management issues.

Things you should consider while Preparing for Chemistry

Build a Strong Foundation

Most of the students skip early chapters and beginning of the lecturers, but they are really important. Basics will help you to make your preparation easy and comfortable. So concentrating on the first chapters is really useful. Even you have no time, go to the introduction chapters and spend time revising them. With the help of those basics, you can easily solve complicated problems.

Know about the Structure

Organic Chemistry is full of compounds and structures. Structures provide properties to a compound which tells the reactions that are possible or not. Understand the formations of structures from a name and hybridization of different atoms, how to identify the number of bonds etc. Mastering organic chemistry will help you solve many tactical questions. If you are perfect at forming the structures, then you can easily solve most of the questions without any worry.

Prepare MindMaps

In organic chemistry, you will find many reagents and reducing agents. Know all the chemical properties of various compounds and classes. Prepare a chart of reactions of specific compounds to get perfect in organic chemistry. You take the example of some compounds and try to solve it yourself so that you will become expert in forming the compounds.

Concentrate on the Problems

If you don’t concentrate on the problems, you can’t get the expected results. Solving organic chemistry requires problem-solving skills and analytical thinking. You must solve many sample papers and mock tests to improve your problem-solving skills.

Understand Rather Than Memorising

Memorizing is not at all a good strategy to score the best marks. Instead, you have to understand each topic clearly. However, there are certain things in Organic Chemistry that are to be memorised like acronyms, functional groups, reagent names, nomenclature terms, functional groups etc. Apart from these topics, you have to understand the remaining topics.

Some Important Topics in Organic Chemistry

Distinction Test

Distinction test is generally asked in between aliphatic compounds and aromatic compounds, compounds with different atoms arrangement and same functional group, compounds with different functional groups.

Conversions

Conversion questions are the mandatory ones and you can easily score in this section. There are many ways to solve this type of questions but you have to find out the most efficient and easiest way to solve these conversions.

Name Reactions

You have to prepare for different reactions. Check for previous question papers and note down the common reactions asked and practice them perfectly. Some of the name reactions are – Clemmensen Reduction reaction, Williamson synthesis, Kolbe’s reaction, Coupling reaction, Hoffmann Bromamide, Sandmeyer reaction, Cannizzaro Reaction etc.

We hope you have found the required information in this article. Stay tuned to get more information on How to Study Organic Chemistry Class 12? Moreover, get details regarding the preparation of every subject and other exams.

The post Step by Step Guide to Prepare Class 12th Organic Chemistry appeared first on Learn CBSE.

Are you worried about scoring the best marks in Physics? Find the one-stop solution for your worry now. We are mentioning the detailed syllabus, solution for How to get 95% in Physics in Class 12? Preparation Tips, books etc. Since class 12 exams are very important for further education process, scoring top marks in all exams is the mandatory thing. Most of the students worry about physics subject. So we are especially concentrating on preparation plan of Class 12 physics subject in this article.

How to get 95% in Physics in Class 12?

Nobody is born genius. Everyone becomes intellectual and clever by knowing several things and practising various topics. Scoring top marks in physics is a dream for most of the students. Good direction and dedication towards preparation will help reach your goal. Even the toppers scoring high in all subjects lack in physics because it is really tough. You need to understand the topics and concepts thoroughly to score best marks in the exam. Focus on your target clearly and concentrate on all topics and formulae. Concentrate more on short answer questions and beat your target.

Preparation Plan for Class 12 Physics

Generally, students feel physics as one such kind of subject where they feel it is tough and difficult to score in this subject compared to other subjects. Also, they put it as the last subject for preparation, but once you know the detailed preparation plan, you can easily score 95% marks in this subject. Here, we are providing with the preparation tips and tricks. Check it out!!

1. Divide the complete syllabus into 3 groups (A, B & C).
2. “A” Group must include all the chapters in which you are perfect and just go through before the exam is required.
3. “B” Group must include all the chapters in which you are little good at and need a revision of 2-3 times.
4.  “C” Group must include all the chapters which you feel as very difficult and require more time to become perfect.
5. Now that you have separated all the chapters into groups, prepare the timetable as per the groups.
6. Allocate less time to A and B groups and more time to C group chapters. As you feel it very difficult, you require more time and concentration for those chapters.
7. If you have any doubts in Group C chapters, clear it then and there by various sources like online videos, lecturers, friends etc.
8. This preparation tips help you to score better marks in the exam as you know all the chapters to concentrate on and easy-going chapters.

Tips to Score Good Marks During the Exam

• Cover all the points in descriptive answers and try to elaborate all the points precisely.
• Attempt all the questions which you know perfectly at the beginning because you can create the first impression on the scrutinizer.
• Use as many graphs and pictorial illustrations to describe your answer perfectly.
• Maintain the answer sheet neatly and do not make so many strikes and underlines.
• For short answer questions, be crisp and point to point.

Class 12 Physics Preparation Plan and Tricks

As mentioned above you have to divide the syllabus and then sit to start your preparation. You have to follow a few steps before starting the preparation.

1. Check your handwriting

Along with your preparation, your handwriting also matters to score 90+ marks in physics subject. Concentrate on the handwriting more because it creates a first impression on the paper. As the scrutinizer can’t see your face, the first impression is the thing that really matters. If you are such a student who wanna be a topper and score high marks, then every single mark holds a value. This is one of the solutions for your worry of How to get 95% in Physics in Class 12?

2. Make your textbooks as Bible

In today’s world, every student only prepares using guides and study materials. But textbooks play a crucial role to cover every single point. Sometimes you find some typical or tricky questions in the exam paper, you can easily solve those questions if you have textbook knowledge. Textbooks will be a good source to improve your knowledge and also to prepare for competitive exams. If you are really afraid to pass in the examination, then go for the guides and complete them and after the completion of the complete syllabus prepare from the reference textbooks.

3. Check your Grammar

Most of the students lose their marks without proper grammar and punctuation. Proper English gives a short boost to your percentage and helps you to score better marks. With proper English Grammar, you can impress the Scrutinizer and score better grades easily. While preparing for the Physics exam, you have to attempt as many mock tests as you can and answer all the questions, so that you will come to know all the mistakes like grammar, punctuations etc.

4. Compete with yourself

Before competing with others, compete with yourself. As we all know, self-motivation is the best motivation. You have to compete with yourself and improve day by day. Keep your target clear to score 90+ marks in physics. As you start practising and attending mock tests, you will come to know your strengths and weaknesses. Each time you attend the mock tests, you have to improve from the previous one and score better marks which helps you to score more marks in the final exam.

5. Don’t worry about the Results

Most of the students worry about results even before attending the exam. Results completely depend on the way to prepare and attend the exam. So, stop worrying about the physics results and make your preparation in such a way that you give your best in the exam.

6. Be Confident! Don’t get Tensed

Get tensed is not the solution to your problem. Indeed you have to attend the worry-free exam. If you feel tensed before the exam, you will lose your concentration and also you will forget some of the answers. Before you go to the physics examination hall, sit quietly for a while, meditate if possible and be confident. Don’t discuss anything with friends or recollect something before entering the examination hall. Stay calm and be confident which ultimately results you to come up with shining colours.

Scoring 90% marks in physics is a daunting task. With good preparation, confidence, strategy and determination, you can easily find the solution fon How to get 95% in Physics in Class 12? Hope you love this article and the above-mentioned tips will guide you in your preparation to score top marks in physics subject.

The post How to get 95% in Physics in Class 12? | Tips to Score 99% Marks appeared first on Learn CBSE.

Exponents are shorthand for repeated multiplication of the same thing by itself. Usually, Exponent says how many times to use a number in the multiplication, In this article, we tried covering everything about the Exponents such as exponential form and product form,  positive and negative rational exponents, negative integral exponents, laws of exponents, etc. in the coming modules.

• Laws of Exponents
• Rational Exponent
• Integral Exponents of a Rational Numbers
• Solved Examples on Exponents
• Practice Test on Exponents
• Worksheet on Exponents

In general, we use exponents to represent large numbers in the short form so that it’s convenient to read. Base a raised to the power n is equal to multiplication of “a” n times. The Process of using Exponents is nothing but raised to the power where the exponent is called power. For better understanding, we even provided solved examples explaining step by step solution.

Example

4*4*4*4*4 can be written as 4⁵  and is read as 4 raised to the power of 5.

In the same way, for any rational number “a”, and a is a positive integer we define aⁿ as a x a x a x a x……a(n times)

(-a)ⁿ = aⁿ  if n is even

= -aⁿ if n is odd

Rational Number a is called the base whereas n is called exponent or power or index. In general, writing the product of writing a rational number by itself multiple times is called exponential notation or power notation.

Examples on Exponential Form

We can denote -3*-3*-3*-3 in the exponent form as (-3)⁴ and is read as -3 raised to the power of 4. in which -3 is the base.

1/2*1/2*1/2 in exponent form as (1/2)³  where 3 is the power and 1/2 is the base.

Examples on Product Form

We can denote (7)³ in the form of 7*7*7 and its product is 343.

Powers with Positive Exponents

For 5² = 5*5 = 25
5³ = 5*5*5 = 125
5⁴ = 5*5*5*5 = 625

Powers with Negative Exponents

Powers with Negative exponents is called Negative Integral Exponents.

Thus, 5^-1 = 1/5

5^-2 = 1/25
5^-3 = 1/125

Therefore, for any non zero rational number and a positive integer, we have a^-n= 1/aⁿ

a^-n is the reciprocal of aⁿ.

Solved Examples on Exponents

1. Express the following in Power Notation

(i) 1/4*1/4*1/4

= (1/4)³

(ii) (-3)*(-3)*(-3)

(-3)³

2. Express the following as Rational Numbers

(i) (2/5)³

= 2/5*2/5*2/5

= 8/125

(ii) (-2)⁴

= -2*-2*-2*-2

= 16/1

3. Express each of the following in the Exponential Form

(i) 625/81

We can write 625 = 5*5*5*5 = 5⁴

81 = 3*3*3*3 = 3⁴

Therefore, 625/81 = 5⁴/3⁴

(ii) -1/32

-1 = -1*-1*-1*-1*-1 = (-1)⁵

32= 2*2*2*2*2 = 2⁵

Therefore, -1/32 expressed in rational number as (-1)⁵/2⁵

4. Express each of the following in product form and find the value.

(i) (1/5)³
= 1/5*1/5*1/5
= 1/125

(ii) (-4)³

= -4*-4*-4

= -64

The post Exponents – Product Form, Exponential Notation, Positive Exponents, Negative Exponents appeared first on Learn CBSE.

Exponents indicate repeated multiplication of a number by itself. There are Six Laws of Exponents in general and we have provided each scenario by considering enough examples. For instance, 5*5*5 can be expressed as 5³. Here 3 indicates the number of times the number 5 is multiplied. Thus, power or exponent indicates how many times a number can be multiplied.

Usually, Exponents abide by certain rules and they are used to simplify expressions and are also called Laws. Let’s dive into the article and learn about the Exponent Laws in detail.

Exponent Rules with Examples

There are six laws of exponents and we have stated each of them by taking examples.

• Product With the Same Bases
• Quotient with Same Bases
• Power Raised to a Power
• Product to a Power
• Quotient to a Power
• Zero Power

Product with Same Bases

In Multiplication of Exponents with Same bases then we need to add the Exponents. We can’t add the Exponents with unlike bases.

According to this law, for any non-zero term a, we have

a^m. aⁿ = a^m+n in which m, n are real numbers.

Example

Simplify 4⁵.4²?

Solution:

Given 4⁵.4²

= 4⁴⁺²

= 4⁶

Simplify (-2)³. (-2)¹?

Solution:

Given (-2)³.(-2)¹

= (-2)³⁺¹

= (-2)⁴

Quotient with Same Bases

In the case of the division with the same bases, we need to subtract the Exponents. According to this rule a^m/aⁿ = a^m-n where a is a non zero integer and m, n are integers.

Example

Find the Value of 10^-4/10^-2?

Solution:

Given 10^-4/10^-2

= 10^-4-(-2)

= 10^-4+2

= 10^-2

= 1/10²

= 1/100

Power Raised to a Power

As per this law, if a is the base and then power raised to the power of base “a” gives the product of powers raised to base “a” such as

(a^m)ⁿ = a^mn where a is a non zero integer and m, n are integers.

Example

Express 16⁴ as a power raised to base 2?

Solution:
We have 2*2*2*2 = 2⁴

Therefore (2⁴)⁴ = 2¹⁶

Product to a Power

According to this rule, for two or more different bases and the same power then

aⁿ. bⁿ = (ab)ⁿ where a is a non zero term and n is an integer.

Example

Simplify and Write the Exponential Form of 1/16*5^-4?

Solution:

We can write 1/16 as 2^-4

= 2^-4*5^-4

= (2*5)^-4

= (10)^-4

Quotient to a Power

According to this law, a fraction of two different bases having the same power is given as

aⁿ/bⁿ = (a/b)ⁿ where a, b are non zero terms and n is an integer.

Example

Simplify the Expression and find the value as 12³/4³?

Solution:

Given Expression is 12³/4³

= (12/4)³

= (3)³

= 27

Zero Power

As per the rule, Any integer raised to the power of 0 is 1 such that a⁰ = 1 and a is a non-zero term.

Example

What is the value of 4⁰ + 11⁰ + 3⁰ + 17⁰ – 3¹?

Solution:

Given 4⁰ + 11⁰ + 3⁰ + 17⁰ – 3¹

Any number raised to the power 0 is 1.

= 1+1+1+1-3

= 4-3

= 1

The post Laws of Exponents | Exponent Rules | Exponents Laws with Examples appeared first on Learn CBSE.

All of you might be aware of Integer Exponents. Let’s get into a little tougher concept i.e. Rational Exponents. Usually, Rational Exponent can be expressed in the form of (b)^m/n where m, n are integers. In Rational Exponents, there are two types namely Positive Rational Exponent and Negative Rational Exponent. Have a glance at the solved examples explaining the concept and get a grip on it and learn how to solve the related problems.

Positive Rational Exponent

Let us consider x and y to be non zero rational numbers and m is a positive integer such that x^m = y then we can express it in the form of x= (y)^1/m. However, we can write y^1/m = m√y and is referred to as the mth root of y.

y^1/3 = 3√y, y^1/5 = 5√y, etc. Consider a positive rational number x having the rational exponent p/q then x can be represented in the following fashion.
X^(p/q) = (x^p)^1/q = q√x^p and is read as qth root of x^p.

X^(p/q) = (x^1/q)^p = (q√x)^p and is read as pth power of qth root of x.

Solved Examples

1. Find the Value of (64)^2/3?

Solution:

= (4³)^2/3
= (4)²
= 16
2. Find the value of (64/27)^5/3?

Solution:

= (64/27)^5/3
= (4³/3³)^5/3
=((4/3)³)^5/3
= (4/3)⁵
= 1024/243

3. Find the value of (256)^1/3?

Solution:

Given (256)^1/3

= (6³)^1/3

= 6

Negative Rational Exponent

If x is a Non- Zero Rational Exponent and m is a positive integer then x^-m = 1/x^m = (1/x)^m i.e. x^-m is the reciprocal of x^m.

The Same Rule is Applicable for Rational Exponents. Consider p/q to be a positive rational number and x > 0 is a rational number.

x^-p/q = 1/x^p/q = (1/x)^p/q i.e. x^-p/q is the reciprocal of x^p/q

If x = a/b then (a/b)^-p/q = (b/a)^p/q

Solved Examples

1. Find 16^-1/2?

Solution:

Given 16^-1/2

= 1/16^1/2
=(1/16)^1/2
=((1/4)²)^1/2

= 1/4

2. Find the value of (32/243)^-4/5?

Solution:

Given (32/243)^-4/5

= 1/(32/243)^4/5
= (243/32)^4/5
= (3⁵/2⁵)^4/5
= ((3/2)⁵)^4/5
= (3/2)⁴
= 81/16

The post Rational Exponent appeared first on Learn CBSE.

In this article of ours, we have discussed Integral Exponents of Rational Numbers. Learn about Positive and Negative Integral Exponents of Rational Numbers in the coming modules. Get to know the Definitions, Solved Examples for understanding the concept better.

Positive Integral Exponent of a Rational Number

Consider a/b as a rational number and n to be a positive integer then

(a/b)ⁿ = a/b*a/b*a/b*a/b…… n times

= (a*a*a*a…..n times)/(b*b*b*b….. n times)
= aⁿ/bⁿ

Therefore, (a/b)ⁿ = aⁿ/bⁿ  and this is applicable for every positive integer “n”.

Example

Evaluate

(i) (3/4)³

Solution:
= (3/4)³

= 3/4*3/4*3/4

= 3*3*3/4*4*4

= 27/64

(ii) (-2/3)⁴

= -2/3*-2/3*-2/3*-2/3

= (-2*-2*-2*-2)/3*3*3*3

= 16/81

(iii) (5/4)⁴

= 5/4*5/4*5/4*5/4

= 5*5*5*5/4*4*4*4
= 625/256

Negative Integral Exponent of a Rational Number

Negative Sign in an Exponent represents the multiplicative inverse or reciprocal. For Negative Exponents, the base shouldn’t be zero as zero doesn’t have a reciprocal. Consider a/b a rational number and n is a positive integer. Then, we can say (a/b)^-n = (b/a)ⁿ

Example

Evaluate

(i) (3/2)^-4

= (2/3)⁴
= 2/3*2/3*2/3*2/3
= (2*2*2*2)/(3*3*3*3)
= 16/81

(ii) (5)^-3

= (1/5)³
=1*1*1/5*5*5

= 1/125

(iii)(1/3)^-3

= (3/1)³

= (3)³
= 27

The post Integral Exponents of Rational Numbers appeared first on Learn CBSE.