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# Class-7 Exponents

Introduction to Exponents

Power of a Negative Rational Number

Law of Exponents

Multiplication of Powers With Same Base

Division of Powers With Same Base

Zero Exponent

Power of a Power

Multiplication of Powers With Same Exponents

Division of Powers With Same Exponents

Negative Exponent

Exponents Test

Exponents Worksheet

## Introduction to Exponents

Many a times we multiply same number multiple times. For example: 2 × 2 × 2 × 2 can be written as 24. Here 2 is known as base and 4 is known as exponents. We can read it as 2 to the power 4 or 2 power 4.

2 × 2 × 2 × 2 = 16 = 2⁴

Let's see some more examples.

Example 1. 3 × 3 × 3 × 3 × 3 = 3⁵ = 243

Example 2. 5 × 5 × 5 × 5 = 5⁴ = 625

Some power has special names, for example if the power of a number is 2 then it is named as square.

4² = 16 (It is read as 4 squared)

Similarly, if the power of a number is 3 then it is name cube.

3³ = 27 (It is read as 3 cubed)

## Power of a Negative Rational Number

If power of a negative rational number is an odd natural number, then the result will be negative. If power of a negative rational number is an even natural number, then the result will be positive. Let's see some examples.

Example 1. Evaluate (−2)³.

Solution. (−2)³ = (−2) × (−2) × (−2) = −8

Example 2. Evaluate (−3)⁴.

Solution. (−3)⁴ = (−3) × (−3) × (−3) × (−3) = 81

Example 3. Evaluate (-23)³. Solution. (-23)³ = (-23) × (-23) × (-23)

{(-2 × -2 × -2)(3×3×3)} = -827

## Law of Exponents

There are 7 laws of exponents, they are mentioned below.

1. Multiplication of powers with same base
2. Division of powers with same base
3. Zero exponent
4. Power of a power
5. Multiplication of powers with same exponents
6. Division of powers with same exponents
7. Negative exponent

## Multiplication of Powers With Same Base

If 'p' is any rational number and a, b are natural numbers, then pa × pb = pa+b. Let's see some examples.

Example 1. Evaluate 23 × 22.

Solution. 23 × 22 = 2 × 2 × 2 × 2 × 2 = 25
In other words, we can write 23 x 22 = 23+2 = 25

Example 2. Evaluate (-2)3 × (-2)2.

Solution. (-2)3 × (-2)2 = {(-2) × (-2) × (-2)} × {(-2) × (-2)} = (-2)5
In other words, we can write (-2)3 x (-2)2 = (-2)3+2 = (-2)5

## Division of Powers With Same Base

If 'p' is any rational number and a, b are natural numbers such that a > b, then pa ÷ pb = pa-b. Let's see some examples.

Example 1. Find the value of 35 ÷ 32.

Solution. 35 ÷ 32 = (3×3×3×3×3)(3×3) = 3 × 3 × 3 = 33

In other words, we can write 35 ÷ 32 = 35-2 = 33

## Zero Exponent

If 'p' is any rational number, then p0 = 1. Let's see some examples.

Example 1. 25 ÷ 25 = (2×2×2×2×2)(2×2×2×2×2) = 3232 = 1

In other words, we can write 25 ÷ 25 = 25-5 = 20 = 1

## Power of a Power

If 'p' is any rational number and 'a', 'b' are natural numbers, then (pa)b = pa×b. Let's see some example.

Example 1. Find the value of (32)3.

Solution. (32)3 = 32 × 32 × 32 = 32+2+2 = 36 = 32×3

So, we can assume that (32)3 = 32×3

## Multiplication of Powers With Same Exponents

If 'p', 'q' are any rational numbers and 'a' is a natural number, then pa x qa = (pq)a. Let's see some example.

Example 1. Find the value of 23 × 33.

Solution. 23 × 33 = 2 × 2 × 2 × 3 × 3 × 3 = (2 × 3) × (2 × 3) × (2 × 3) = (2 × 3)3

## Division of Powers With Same Exponents

If 'p', 'q' (q ≠ 0) are any rational numbers and 'a' is a natural number, then pa ÷ qa = (pq)a.

Example 1. Find the value of 23 ÷ 53.

Solution. 23 ÷ 53 = (2×2×2)(5×5×5) = 25 × 25 × 25 = (25)3

Example 2. Find the value of (-2)3 ÷ 73.

Solution. (-2)3 ÷ 73 = (-2 × -2 × -2)(7 × 7 × 7) = -27 × -27 × -27 = (-27)3

## Negative Exponent

If 'p' is any non-zero rational number and 'n' is any natural number, then p-n= 1pn.

1pn = p0pn = p0-n = p-n
Hence it is proved that p-n= 1pn

Exponents - 1

Exponents - 2

## Class-7 Exponents Worksheet

Exponents Worksheet - 1

Exponents Worksheet - 2

Exponents Worksheet - 3