# Class-7 Exponents

Power of a Negative Rational Number

Multiplication of Powers With Same Base

Division of Powers With Same Base

Multiplication of Powers With Same Exponents

Division of Powers With Same Exponents

## 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 (^{-2}⁄_{3})³.
**Solution.** (^{-2}⁄_{3})³ = (^{-2}⁄_{3}) × (^{-2}⁄_{3}) × (^{-2}⁄_{3})

{^{(-2 × -2 × -2)}⁄_{(3×3×3)}} = ^{-8}⁄_{27}

## Law of Exponents

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

- 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

## Multiplication of Powers With Same Base

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

**Example 1.** 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}

**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 *p ^{a} ÷ p^{b} = p^{a-b}*. Let's see some examples.

**Example 1.** Find the value of 3^{5} ÷ 3^{2}.

**Solution.** 3^{5} ÷ 3^{2} = ^{(3×3×3×3×3)}⁄_{(3×3)} = 3 × 3 × 3 = 3^{3}

In other words, we can write 3^{5} ÷ 3^{2} = 3^{5-2} = 3^{3}

## Zero Exponent

If *'p'* is any rational number, then *p ^{0} = 1*. Let's see some examples.

**Example 1.** 2^{5} ÷ 2^{5} = ^{(2×2×2×2×2)}⁄_{(2×2×2×2×2)} = ^{32}⁄_{32} = 1

In other words, we can write 2^{5} ÷ 2^{5} = 2^{5-5} = 2^{0} = 1

## Power of a Power

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

**Example 1.** Find the value of (3^{2})^{3}.

**Solution.** (3^{2})^{3} = 3^{2} × 3^{2} × 3^{2} = 3^{2+2+2} = 3^{6} = 3^{2×3}

So, we can assume that (3^{2})^{3} = 3^{2×3}

## Multiplication of Powers With Same Exponents

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

**Example 1.** Find the value of 2^{3} × 3^{3}.

**Solution.** 2^{3} × 3^{3} = 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 p^{a} ÷ q^{a} = (^{p}⁄_{q})^{a}.

**Example 1.** Find the value of 2^{3} ÷ 5^{3}.

**Solution.** 2^{3} ÷ 5^{3} = ^{(2×2×2)}⁄_{(5×5×5)} = ^{2}⁄_{5} × ^{2}⁄_{5} × ^{2}⁄_{5} = (^{2}⁄_{5})^{3}

**Example 2.** Find the value of (-2)^{3} ÷ 7^{3}.

**Solution.** (-2)^{3} ÷ 7^{3} = ^{(-2 × -2 × -2)}⁄_{(7 × 7 × 7)} = ^{-2}⁄_{7} × ^{-2}⁄_{7} × ^{-2}⁄_{7} = (^{-2}⁄_{7})^{3}

## Negative Exponent

If *'p'* is any non-zero rational number and *'n'* is any natural number, then *p ^{-n}*=

^{1}⁄

_{pn}.

^{1}⁄

_{pn}=

^{p0}⁄

_{pn}= p

^{0-n}= p

^{-n}

Hence it is proved that p

^{-n}=

^{1}⁄

_{pn}

## Class-7 Exponents Test

## Class-7 Exponents Worksheet

## Answer Sheet

**Exponents-Answer**Download the pdf

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