Exponent Definition

An exponent is the number of times that a certain number is multiplied by itself. It is written as a small-sized number at the top right of the number.

In the example above, 5^2 is read as "5 to the power of 2" or "5 raised to 2".

Product form of numbers

Large numbers broken down into smaller forms is known as product form of numbers.

To find the product form of numbers let's say 47
47 is written in the product form as 47= 4 × 4 × 4 × 4 × 4 × 4 × 4

Expressing the product form of numbers by exponents

To express product from of the numbers by exponents we count that how many times the number multiplied by itself. That is the exponent of that number.

To find the exponent form of numbers let's say 32.
Write the product form of 32. 32= 2 x 2 x 2 x 2 x 2= 25.
Here 2 is multiplied by 5 times with itself. So that exponent is 5 and base is given 2.
We get the exponent form is 25.

1. Express 128 as a power 2.

Explanation:

Let us find the prime factors of 128.
128= 2 x 2 x 2 x 2 x 2 x 2 x 2 = 27
Here 2 is multiplied by 7 times with itself. So that exponent is 7 and base is given 2.
We get the exponent form is 27.

2. Express 32 as a power 2.

Explanation:

Let us find the prime factors of 32.
32= 2 x 2 x 2 x 2= 24
Here 2 is multiplied by 5 times with itself. So that exponent is 4 and base is given 2.
We get the exponent form is 24.

Square of a Number

A square of number is the product of some integer with itself.
For example, 25 is a square number, since it can be written as 5 × 5.

Note:

•  To find a square of number we multiply the number two times by itself.
•  For example square of 4 can be represented as "4×4=16". We can express it in exponent form as 4^2.
•  Square of negative number is positive number.

1. Find the square of 35.

Explanation:

Square of 35.
= (-35)2
= (-35) × (-35)
= 1225

Examples square of variables:

• 1. ab2 = a × b × b
• 2. (ab)2 = ab × ab
• 3. (ab)-2 = -(ab) × (-ab)= (ab)2

Cube of a Number

A cube of number is just the number multiplied by 3 times with itself.

For example, to find cube of 3.
Cube of 3 is 3 times 3.
Cube of 3 = 3 × 3 × 3= 27.
We can express it in exponent form as 33

Note:

•  Cube of negative number is negative number.

1. Find the cube of 25.

Explanation:

Find the cube of 25.
= (-25)3
= (-25) × (-25) × (-25)
= -15625

Examples cube of variables:
• 1. ab3= a × b × b x b
• 2. (ab)3= ab×ab×ab

What is the value of numbers in exponent form?

We can find the value of numbers given in exponent form by multiplying the base with itself as many times the exponent is given.

To find the value of 35.
35 mean multiplying the 3 with itself 5 times.
35= 3 × 3 × 3 × 3 × 3
3 × 3 × 3 × 3 × 3= 243.

Box Arrangement Examples

Please see the picture below. Boxes are in group of 4. Count the group number of boxes and write them in exponent & product form. How could you read this number? Find the value for this number in exponent form.

Explanation :

The group numbers of boxes are 3. In every group there are 4 boxes. So that base is 4 and exponent is 3. We can write this in exponent form as 43.
The product form of 43= 4×4×4
We can read 43 as 4 raised to 3.
The value of 43= 4×4×4= 64

Star Eaxmple

Please, see the picture below. Stars are in group of 12. Count the group number of stars and write them in exponent & product form. How could you read this number? Find the value for this number in exponent form.

Explanation :

The group numbers of stars are 4. In every group there are 12 stars. So that base is 12 and exponent is 4. We can write this in exponent form as 124.
The product form of 124= 12 × 12 × 12 × 12
We can read 124 as 12 raised to 4.
The value of 124= 12 × 12 × 12 × 12= 20736

Exponent rules

Any number with an exponent of 0 (except 0) is equals to 1. So that x0 = 1.
Examples: 50= 1, 150= 1, 1230= 1, 1540= 1, 1200= 1

When base raised to the positive power value; x1= x
Examples: (10)1= 10, (15)1= 15, (124)1= 124 and (-10)1= -10, (-15)1= -15, (-124)1= -124

Simple Example

Explanation:

(2/3)6
Here 2/3 is the base and 6 is exponent.
For (2/3)6 mean 2/3 is 6 times. So that,
(2/3)6= 2/3× 2/3 × 2/3× 2/3 × 2/3 ×2/3= 2 × 2 × 2 × 2 × 2 × 2 / 3 × 3 × 3 × 3 × 3 × 3= 64/729

When base raised to the negative power value; x-1 = 1/x.
Examples: 5-6= 1/56, 4-3= 1/43, 7-5= 1/75

2. What is the value of (12)-4

Explanation:

(12)-4
Here 12 is the base and -4 is exponent.
For (12)-4mean 1/ (12)4
(12)-4=1/ (12)4
=1/12 × 1/12 × 1/12 × 1/12= 1 × 1 × 1 × 1 / 12 × 12 × 12 × 12 = 1/ 20736

Cube Example

A cube shaped container is one third full with kerosene oil. Each side of the container is 9 inches long. What is the volume of kerosene in cubic inches?

Explanation :

Volume of a cube= (side)3
Volume of a cube container = 93= 9×9×9= 729.
Volume of a cube container is 729 cubic inches.
The volume of kerosene in cubic inches
= 729 /3= 243 cubic inches

Rule for multiplication when the base is the same

When we multiply the numbers having the same base, the base of the product remains the same and exponents are added to get the index of the product.

For example: 84× 85= 84+5=89

Note:

•  If a is any rational number and m and n are any positive integers then am× an = am + n

Simple Example

Explanation:

75×74
= (7 × 7 × 7 × 7 × 7)×( 7× 7 × 7 × 7)
= (7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7)
= the number 7, (5 + 4) times=79
So that, 75×74=79

(7/5)3×(7/5)8
= (7/5)3+8
= (7/5)11

(y)4×(y)6
= (y)4+6
= (y)10

Project Completion Example

A group of employers were given a project has three parts. There are 32 ways to complete the 2 documents in Part A. There are 32 ways complete the 2 documents in Part B. How many ways are there to complete the project all 4 documents?

There are 81 ways to complete the project.

Explanation :

There are 32 ways to complete the 2 documents in Part A. There are 32 ways complete the 2 documents in Part B. The number of ways to complete the project all 4 documents is the product of 32and 32
The product of 32and 32
= 332
= 32+2
= 34
= 81

Rule for division when the base is the same

If a is any rational number other than 0 and m and n are any positive integers such that m>n, then am÷ an = am-n

Note:

•  When we divide the numbers having the same base, the base remains the same and exponents are subtracted.
•  When the index of the dividend is greater than that of divisor, subtract the exponent of the divisor from that of the dividend.

= 89-4
= 85
84÷ 89
= 1/89-4
= 1/85

1. 68÷64?

Explanation:

68÷64
= (6×6×6×6×6×6×6×6) ÷ ( 6×6×6×6)
= ( 6×6×6×6)
= 1296

1. (2/3) 8÷ (2/3)3?

Explanation:

= (2/3)8-3
=(2/3)5
= (2/3)× (2/3)× (2/3)× (2/3)× (2/3)
= 2×2×2×2×2 / 3×3×3×3×3
= 32/243

(a) 4×(a) 6
=1/(a) 6-4
=1/(a) 2

Survey Based Example

There is a survey taken in society about working from home jobs for women and man. It is found that 76 women were doing jobs from home but only 72 men were doing jobs from home. How many times more women were doing jobs from home?

Explanation :

To finding more women were doing jobs from home, we have to divide these two expressions shown for women and men. That is 76 and 72
76/72
= 76-2
= 74
=7×7×7×7
= 2401

Rule for index of a number in index form

If x is any rational number and m and n are positive integers then
(xm )n= (x)m × n

If we say, (23 )5
(23 )5
Multiply the index.
(23 )5= (2)3 × 5= 215

(a4 )5
(a4 )5
= a4 × 5
= a20

1. Find the value [(-3/4)2]2

Explanation:

[(-3/4)2]2
We will multiply the index numbers.
= (-3/4)2×2
= (-3/4)4
= -3× -3× -3× -3/ 4× 4 × 4× 4
= 81/256

Rule for the index of a product

If x and y are any rational number and m is any positive integers then
(x× y) m = xm × ym .

If we say, (4×3)7.
(4×3)7= (4)7× (3)7

Simple Example

(a×b)5
(a×b)5 = (a)5× (b)5

1. Find the value (5×2)3

Explanation:

(5×2)3
= (5)3× (2)3
= (5×5×5) ×(2×2×2)
= 1000

1. Find the value ( 1/3× 2/5 ) 2

Explanation:

( 1/3× 2/5 ) 2
= (1/3)2 × (2/5 )2
= 1×1×2×2/ 3×3×5×5
= 4/225

Rule for index of a number in the numerator-denominator form

If x / y is any rational number and m is any positive integers then
(x/ y) m = xm/ ym

Let us say,
(3/ 5) 2 = 32/ 52 and (x/y) 2 = x2/ y2

1. Find the value (5/2)4

Explanation:

(5/2)4
= (5)4/ (2)4
= (5×5×5×5) / (2×2×2×2)
= 625/16

1. Find the value (-6/5)2

Explanation:

(-6/5)2
= (-6)2/ (5)2
= (-6×-6) / (5×5)
= 36/25

1. Find the value (5×2)3 ÷ (5×2)1

Explanation:

(5 × 2)3 ÷ (5 × 2)1
We will first subtract the index here.
= (5×2) 3-1
= (5×2)2
= (5)2× (2)2
= 5 × 5 × 2 × 2
= 100

Explanation:

We will first add the index here.
= (51 )2+3
= (51 )5
= 51×5
=5 5
= 3125

3. Evaluate the expression (a × b) 3 ÷ (a × b) 5 for a=2 and b = 3

Explanation:

(a × b) 3 ÷ (a× b) 5
= 1/ (a × b) 5-3
= 1/ (a × b) 2
= 1/ (2 × 3) 2 put the value for a=2 and b = 3
= 1/ (2)2× (3)2
1/ 2 × 2 × 3 × 3
=1/36

4.Evaluate the expression (a) 0 ÷ (b) 0 for a=15 and b = 17

Explanation:

(a) 0 ÷ (b) 0
= (15) 0 ÷ (17) 0 put the value for a=15 and b = 17
= 1÷ 1
=1

Fractional index

The square of number is shown by writing the number with exponent 2.
Index ½ represents the square root of the number.
These are known as fractional indices.

Let us say square of 4, we will write as 42.
Now, if we want to write square root of 4, we will write as 41/2.
As well as the cube root of 8 is written as 81/3

Note:

• The nth root of a number is shown by the index 1/n.

How to read the numbers as fractional indices?

See the following examples:

91/2 means the square root of 9.
10001/3 means the cube root of 1000.
6251/4 means the fourth root of 625.
10001/10 means the square root of 1000.

1. What is the fourth root of 256?

Explanation:

4 × 4 × 4 × 4 = 256.
Hence, 44 = 256.
Conversely, 4 is the fourth root of 256.

1. Find sqaure root of 256.

Explanation:

16 × 16 = 256.
Hence, 162 = 256.
Conversely, 16 is the square root of 256.

Medium Example:

Find the values of the following index numbers.

•  1251/3= Cube root of 125 = 5
•   1441/2= Square root of 144 = 12
•   2161/3= Cube root of 216 = 6
•   1691/2= Square root of 169 = 13

Multiplication fractional indices

As you know If a is any rational number and m and n are any positive integers then
am× an = am+n
The same rule is applied for fractional indices when we multiply the numbers as index form with the same base.

If we say, 71/2× 71/3
Add the index numbers. 71/2× 71/3 = 71/2+ 1/3
Simplify added index form. 71/2+ 1/3 = 73+2/6
73+2/6= 75/6 means the 5th power of the 6th root of 7.

91/3× 91/3
= 91/3+ 1/3
= 91+1/3
= 92/3

141/4× 141/5
= 141/4+ 1/5
= 145+4/20
= 149/20

x1/3× x1/5
= x1/3+ 1/5
= x5+3/15
= x8/15

Division fractional indices

If a is any rational number other than 0 and m and n are any positive integers such that m>n, then am÷ an = am-n

The same rule is applied for fractional indices when we multiply the numbers as index form with the same base.
If we say,
71/3÷ 71/3

Subtract the index numbers. 71/3÷ 71/3= 71/3 - 1/3
Simplify index form. = 71/3 - 1/3 = 70
Write the value. 70 = 1

1. 131/4÷ 131/5

Explanation:

131/4÷ 131/5
= 131/4- 1/5
= 135-4/20
= 131/20
131/20 means the 20th root of 13.

m1/3÷ m1/5
= m1/3 - 1/5
= m5-3/15
= m2/15

81/4÷ 81/6
= 81/4 - 1/6
= 8 3-2/12
= 8 1/12

Rule for index of a number for fractional indices

If x is any rational number and m and n are positive integers then
(xm )n= xm × n

If we say, (21/3 )1/5
Multiply the index numbers. (21/3 )1/5 = 21/3 ×1/5
Simplify index form. 21/3 ×1/5 = 21/15
Write the value. 21/15

1. Simplify [(25)1/2]1/2

Explanation:

(25) 1/2 ×1/2

Multiply the index numbers.
= 251/4

2. Simplify [ (8)1/2]1/9

Explanation:

8 1/2 ×1/9
Multiply the index numbers.
= 81/18

3. Simplify [(y)1/2]1/3

Explanation:

y 1/2 × 1/3
Multiply the index numbers.
= y1/6

Rule for the index of a product for fractional indices

If x and y are any rational number and m is any positive integers then
(x× y) m = xm × ym .

We can say,
(5×3)1/7= (5) 1/7× (3) 1/7
Or (a×b) 1/5= (a) 1/5× (b) 1/5

Simple Example

Simplify (5 × 9) 1/3
(5×9) 1/3
= (5) 1/3× (9) 1/3

1. Find the value (25 × 8) 1/3

Explanation:

(25×8) 1/3
= (25) 1/3× (8) 1/3
= 5 × 2
= 10

2. Find the value (100 × 64) 1/2

Explanation:

(100 × 64) 1/2
= (100) 1/2× (64) 1/2
= 10 × 8
= 80

1. Find the value (625 × 256) 1/2÷ (625 × 256) 1/4

Explanation:

(625 × 256) 1/2 ÷ (625 × 256) 1/4
= (625 × 256) 1/2 - 1/4
= (625 × 256) 1/4
= (625) 1/4 × (256) 1/4
= 5 × 4
= 20

2. Find the value [(81)1 ]1/2 × [(81)1 ]1/4

Explanation:

(625 × 256) 1/2 ÷ (625 × 256) 1/4
[(81)1 ]1/2 × [(81)1 ]1/4
= [(81)1 ]1/2 + 1/4
= (81 )3/4
Means the 3nd power of the 4th root of 81.
= 27

Evaluate the expression (a× b) 1/3 ÷ (a× b) 1/5 for a=2 and b = 3

Explanation:

(a× b) 1/3 ÷ (a× b) 1/5 put the value for a=2 and b = 3
= (2× 3) 1/3 ÷ (2× 3) 1/5
= (2× 3) 1/3- 1/5
= (2× 3) 5-3/15
= (2× 3) 2/15
= 22/15× 3 2/15

Points to Remember

For any given number make pairs...

Quick Tips

•  When we multiply/divide the Indices having same base, the base remains as it is.
•  To write the numbers as a power of 10, the first number must between 1 and 10 and the other is power of 10.

Think:

•  Which one is greater 73 or 37?
•  What could be the value of (1)1000
•  If 83÷ 32 is given. Can you add the exponents? If No! ‘Why’?

Worksheets

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