Write Exponents and Standard Index Form with Examples

What are exponents?

A longer number which can be written its same factors e.g. if x is multiplied n times by itself, it would look like
a × a × a × a × a × a × a × a ×…. n times, where a is an integer. To short such numbers they can be written in exponent form like aⁿ.
Here, aⁿ is pronounced as a raised to power n or nth power of a.
x is called base and n is the power or the exponent.

Example of exponent

2 is multiplied 5 times
It can be written as 2 × 2 × 2 × 2 × 2 × = 32
Its exponent form can be written as 2⁵ and read as 2 raised to power 5 or 5th power of 2
In 2⁵, 2 is the base and 5 is the power or exponent.

Base should not be zero in the exponent form.

Next are the rules to change any number into its exponent form, these rules are also known as Laws of Exponents.

Laws of exponents

There are six important laws of exponents those help to solve problems of exponents when their bases or powers are multiplied or divided.

1. When same bases are multiplied

a^m × aⁿ is such case where bases are same i.e. a, where a ≠ 0.
As per this rule, if same bases, which is a, are multiplied, the powers m and n will get added up and the new common base will be again a
i.e. new power will be m + n and the base will be a, which can be written as a^{m + n}
or a^m × aⁿ = a^{m + n}

Example of same bases multiplied

2³ × 2²
= 2^{3 + 2}
= 2⁵

2. When same bases are divided

^{a^m}_aⁿ or a^m ÷ aⁿ is an example where same bases a are divided.
According this rule, if same bases are divided then powers m and n will be subtracted.
i.e. base a will have new power will m – n, which can be written as a^{m – n}
∴ ^{a^m}_aⁿ = a^{m – n}

Example of same bases divided

^2⁶_2⁴
= 2^{6 – 4}
= 2²

3. When exponent is 0

When exponent of a number is 0, then that becomes equal to only 1.
a⁰ = 1, where a ≠ 0

Examples of 0 exponent

Example 1: 2⁰
= 1

Example 2: ^2⁴_2⁴
= 2^{4 – 4}
= 2⁰
= 1

4. Power of a power

When a power has its own power, then the powers get multiplied.
(a^m)ⁿ = a^{m × n} = a^mn, where a ≠ 0

Example of power of a power

(5²)³
= 5^{2 × 3}
= 5⁶

5. Product of bases with same exponents

When the numbers with different bases and same exponents are multiplied, then a common exponent is set on the multiplication value of the bases.
a^m × b^m = (a × b)^m = (ab)^m

Example of same exponent's bases product

3⁵ × 2⁵
= (3 × 2)⁵
= 6⁵

6. Division of bases with same exponents

In the division of numbers with different bases and same exponents, then the two numbers are divided and a common exponent is set on the division of the bases. ^{a^m}_b^m
= (^a_b)^m

Example of division of bases with same exponents

^4⁵_6⁵
= (⁴₆)⁵

What is standard index form?

Standard Index Form or Standard Form (in United Kingdom) or Scientific Notation is a way of writing the long and short numbers using exponents or powers.

Long and small numbers can be written in the form of exponents to make them short and readable. Any number which is written in the form of k × 10ⁿ, where k is the natural or a decimal number with ones place only, is the standard index form.

Example of standard form to understand

Standard form of long number
625481453588462135 is a long number.
It can be written as 6.25 × 10¹⁷.

Standard form of small number
0.00000005 is an example of small number.
0.00000005 can be written as 5 ×10^-8.

There are actual scientific constant values which are also written in standard forms are listed in tables below.
Table: list of examples of exponent forms of long numbers.

Name	Standard Index Form
Mass of earth	5.9722 × 10²⁴ kg
Distance of sun from earth	1.496 × 10⁸Km
Speed of light	3 × 10⁸ m/s

Table: list of examples of exponent forms of short numbers.

Name	Standard Index Form
Mass of electron	9.109 × 10^-31 kg
Mass of proton	1.672 × 10^-27 kg
Mass of neutron	1.674 × 10^-11 kg

How to write standard index form?

Check whether the given number is less than 1 or between 1 and 10 or greater than 10.

1. If number less than 1

If number is less than 1, then move the decimal point to the right until there is only one digit on left side of the decimal point.
Count the number of decimal places shifted to right
Then write its product with 10^-n Here, n is the number of places the decimal point has shifted to right.

Example of standard index form, if number less than 1

Example 1: Write 0.327 in standard index form.
0.327 is less than 1
So shift the decimal point to the right side to make the number upto ones place only.
i.e. if decimal is shifted one place to the right then it becomes 3.27
Count the number of digits the decimal point moved to right which is 1
So write the product of 3.27 and 10 with a negative power of 1 or 10^-1
∴ Standard index form of 0.327 is 3.27 × 10^-1

Example 2: Write 0.0005467 in standard index form.
0.0005467 is less than 1
Shift decimal point to right by 4 places so that 5 comes at ones place i.e. 5.467
So write the product of 5.467 and 10 with a negative power of 4 or 10^-4
∴ Standard index form of 0.0005467 is 5.467 × 10^-4

2. If number lies from 1 to 10

If number is equal to 1 or between 1 and 10, then product of the given number and 10⁰ is written to make its standard form.

Example of standard index form, for numbers from 1 to 10

Write 4 in standard index form.
4 lies between 1 and 10
So write the product of 4 and 10 with power of 0 or 10⁰
∴ Standard index form of 4 is 4 × 10⁰

3. If number greater than 10

When the numbers are very large and are greater than 10, then move decimal point to the left side upto the digit so that there left only one digit on the left of decimal point.
Write the product of the above number and 10ⁿ, where n is counted as the number of places the decimal point has moved to left.

Example of standard index form, if number greater than 10

Example 1: Write 17.52 in standard index form.
17.52 is greater than 1
So shift decimal point to left by 1 place to make the number upto ones place i.e. 1.752
Write the product of 1.752 and 10 with a positive power of 1 or 10¹
∴ Standard index form of 17.52 is 1.752 × 10¹

Example 2: Write 321500 in standard form.
321500 is greater than 1
So shift decimal point to left by 5 places to make the number upto ones place i.e. 3.21500
Write the product of 3.21500 and 10 with a positive power of 5 or 10⁵
∴ Standard index form of 321.500 is 3.21500 × 10⁵
or 3.21500 × 10⁵ can be written as 3.215 × 10⁵

Solved Examples

1) Write the following in exponential form.

10 × 10 × 10
12 × 12 × 12 × 12
21 × 21 × 21 × 21 × 21
100 × 100

Solutions:

10 × 10 × 10
= 10³
12 × 12 × 12 × 12
= 12⁴
21 × 21 × 21 × 21 × 21
= 21⁵
100 × 100
= 100²

2) Find value of the following.

5²
14³
(-3)⁴
2⁵
100⁰
(²₃)³
4² × 5²
-4 × (-5)³
9² × 1³
10² × 0

Solutions:

5²
= 5 × 5
= 25
14³
= 14 × 14 × 14
= 2744
(-3)⁴
= (-3) × (-3) × (-3) × (-3)
= 81
2⁵
= 2 × 2 × 2 × 2 × 2
= 32
100⁰
= 1
(²₃)³
= ²₃ × ²₃ × ²₃
4² × 5²
= 4 × 4 × 5 × 5
= 16 × 25
= 400
-4 × (-5)³
= -4 × (-5) × (-5) × (-5)
= -4 × (-125)
= 500
9² × 1³
= 9 × 9 × 1 × 1 × 1
= 81
10² × 0
= 10 × 10 × 0
= 0

3) Write the following numbers in exponent form.

9
144
81
625
1000
⁹₄
²¹⁶₁₂₅
¹³³¹₁₆₉
- ³⁴³₇₂₉
- 32

Solutions:

9
= 3 × 3
= 3²
144
= 12 × 12
= 12²
81
= 3 × 3 × 3 × 3
= 3⁴
625
= 5 × 5 × 5 × 5
= 5⁴
1000
= 10 × 10
= 10³
⁹₄
= ³₂ × ³₂
= ^3²_2²
= (³₂)²
²¹⁶₁₂₅
= ^{6 × 6 × 6}_{5 × 5 × 5}
= ^6³_5³
= (⁶₅)³
¹³³¹₁₆₉
= ^{11 × 11 × 11}_{13 × 13}
= ^11³_13²
= ^(11)³_(13)²
- ³⁴³₇₂₉
= - ^{7 × 7 × 7}_{9 × 9 × 9}
= - ^7³_9³
= - ^(7)³_(9)³
- 32
= - 2 × 2 × 2 × 2 × 2
= (- 2)⁵

4) Simplify the following by using the laws of exponents.

5³ ÷ 5²
(7^2)⁴
4¹ × 4² × 4³
11² × 9²
(3² × 3⁴) ÷ 3⁵
2^-2 × 2^-3
(⁵₃)^-2 × (³₅)^-3
(2^-2 × 3^-2)^-2 × 6^-2
100⁰ × 100¹ × 100² × 100³
2² × 5² ÷ 10²

Solutions:

5³ ÷ 5²
= 5^{3 - 2}
= 5¹
= 5
(7^2)⁴
= 7^{2 × 4}
= 7⁸
4¹ × 4² × 4³
= 4^{1 + 2 + 3}
= 4⁶
11² × 9²
= 11 × 9²
= 99²
(3² × 3⁴) ÷ 3⁵
= 3^{2 + 4} ÷ 3⁵
= 3⁶ ÷ 3⁵
= 3^{6 - 5}
= 3¹
= 3
2^-2 × 2^-3
= 3^{2 + 4} ÷ 3⁵
= 3⁶ ÷ 3⁵
= 3^{6 - 5}
= 3¹
= 3
(⁵₃)^-2 × (³₅)^-3
= ¹ _(⁵₃)² × ¹ _(³₅)³
= ¹ _{^5²_3²} × ¹ _{^3³_5³}
= ^3²_5² × ^5³_3³
= ⁵₃
(2^-2 × 3^-2)^-2 × 6^-2
= (¹_2² × ¹_3²) × 6^-2
= ¹_{(2 × 3)²} × 6^-2
= ¹_6² × 6^-2
= ¹_6² × ¹_6²
= ¹_6⁴
100⁰ × 100¹ × 100² × 100³
= 100^{0 + 1 + 2 + 3}
= 100⁶
2² × 5² ÷ 10²
= (2 × 5)² ÷ 10²
= 10² ÷ 10²
= 10^{2 - 2}
= 10⁰
= 1

5) Find value of x for 2² × 2³ = 2^x exponential form.

2² × 2³ = 2^x
2^{2 + 3} = 2^x
2⁵ = 2^x
By equating exponents on both sides
5 = x
or x = 5

6) Find value of y for 3⁶ × 3⁴ = (3²)^x

3⁶ × 3⁴ = (3²)^x
3^{6 + 4} = 3^2x
3¹⁰ = 3^2x
By equating exponents on both sides
10 = 2x
x = ¹⁰₂
x = 5

7) Find value of m for 5^{m + 2} × 5^{m - 1} = 125

5^{m + 2} × 5^{m - 1} = 125
5^{m + 2 + m - 1} = (5 × 5 × 5)
5^{2m + 1} = 5³
By equating exponents on both sides
2m + 1 = 3
2m = 3 - 1
2m = 2
m = ²₂
m = 1

8) Find value of n for (7²)ⁿ ÷ 7² = 2401

(7²)ⁿ ÷ 7² = 2401
7²ⁿ ÷ 7² = (7 × 7 × 7 × 7)
7^{2n - 2} = 7⁴
By equating exponents on both sides
2n - 2 = 4
2n = 4 + 2
2n = 6
n = ⁶₂
n = 3

9) Find value of t in (²₇)² × (²₇)⁴ = (²₇)^{t - 1}

(²₇)² × (²₇)⁴ = (²₇)^{t - 1}
(²₇)^{2 + 4} = (²₇)^{t - 1}
(²₇)⁶ = (²₇)^{t - 1}
By equating exponents on both sides
6 = t - 1
t = 6 + 1
t = 7

10) Find value of x in (³₁₁)^-2 × (³₁₁)^-3 = (³₁₁)^5x

(³₁₁)^-2 × (³₁₁)^-3 = (³₁₁)^5x
(³₁₁)^{-2 + (-3)} = (³₁₁)^5x
(³₁₁)^{-2 - 3} = (³₁₁)^5x
(³₁₁)^-5 = (³₁₁)^5x
By equating exponents on both sides
-5 = 5x
x = - ⁵₅
x = - 1

Write True or False Worksheet

Type:	True False
Count:	1

S.N.	Statement	✓ or ✕
1)	The value of 1⁰ is 1.
2)	Cube of 2 is 8.
3)	2² × 2^-2 gives 1.
4)	The square of 4 is 8.
5)	a⁵ × a^-7 is equal to a².
6)	(²₃)⁰ gives the value of ²₃.
7)	For 11^x = 121, the value of x is 2.
8)	(¹⁴₅)² is equal to (¹₅)⁸
9)	1000 cm can be written in exponent form (10)²
10)	(5)^-3 gives the value of ¹₁₂₅

True False

PDF Worksheet

Match Columns Worksheet

Type:	Matching
Count:	1

1)	12 × 12	a)	216
2)	(-2)³ × (-3)³	b)	1
3)	(4)¹² ÷ (4)⁸	c)	21
4)	20⁰ × 10⁰	d)	25
5)	(-5)²	e)	144
6)	(21)⁵ × (21)^-3 ÷ (21)	f)	256

Matching

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Solve Questions Worksheets

Type:	Solve Questions
Count:	2

Solve the following questions.

S.N.		Your answer
1)	2¹³ × 2⁴	_______
2)	3² × 3^-7	_______
3)	4¹² ÷ 4⁸	_______
4)	5³ ÷ 125	_______
5)	343 ÷ 7²	_______
6)	(⁵₃)² × (⁵₃)^-2	_______
7)	3⁴ × 5⁴	_______
8)	(10²)³	_______
9)	(81)⁴ ÷ 9²	_______
10)	(⁷₂)⁴ ÷ (²₇)⁴	_______

Solve Questions

PDF Worksheet

Express the following numbers into standard index form or scientific notation.

S.N.		Your answer
1)	0.32570	_______
2)	535458	_______
3)	4.83978	_______
4)	0.078421	_______
5)	0.0004967	_______
6)	95168	_______
7)	99800215	_______
8)	38400000	_______
9)	684000	_______
10)	786540	_______