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

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 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 p^a ÷ p^b = p^a-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 = 3^5-2 = 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)} = ³²⁄₃₂ = 1

In other words, we can write 2⁵ ÷ 2⁵ = 2^5-5 = 2⁰ = 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 (3²)³.

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

So, we can assume that (3²)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 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 p^a ÷ q^a = (^p⁄_q)^a.

Example 1. Find the value of 2³ ÷ 5³.

Solution. 2³ ÷ 5³ = ^(2×2×2)⁄_(5×5×5) = ²⁄₅ × ²⁄₅ × ²⁄₅ = (²⁄₅)³

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

Solution. (-2)³ ÷ 7³ = ^{(-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