Indices

Laws of Indices

Lesson Objectives

Understand what indices (exponents) represent.
Apply the basic laws of indices: product, quotient, power of a power, zero, and negative indices.
Simplify algebraic expressions using index laws.
Evaluate numerical expressions with indices.

Introduction

Indices, or exponents, represent repeated multiplication. For example, $2^3$ means $2 \times 2 \times 2 = 8$ . Understanding the laws of indices helps simplify complex expressions and solve equations efficiently.

Core Lesson Content

Product of Powers Rule

$a^m \times a^n = a^{m+n}$

Example 1a (Easy): Simplify

4^2 \times 4^3

= 4^{2+3} = 4^5 = 1024

Example 1b (Difficult): Simplify

x^2 \times x^5 \times x^{-3}

= x^{2+5-3} = x^4

Quotient of Powers Rule

$a^m \div a^n = a^{m-n}$

Example 2a (Easy): Simplify

9^5 \div 9^2

= 9^{5-2} = 9^3 = 729

Example 2b (Difficult): Simplify

a^6 \div (a^2 \times a^{-1})

= a^6 \div a^{2-1} = a^6 \div a^1 = a^5

Power of a Power Rule

$(a^m)^n = a^{m \times n}$

Example 3a (Easy): Simplify

(2^3)^2

= 2^{3 \times 2} = 2^6 = 64

Example 3b (Difficult): Simplify

(x^2 y^3)^4

= x^8 y^{12}

Zero Exponent Rule

$a^0 = 1$ for any $a \ne 0$

Example 4a (Easy): Evaluate

7^0

= 1

Example 4b (Difficult): Evaluate

(2x^3 y^5)^0

= 1

Negative Exponent Rule

$a^{-m} = \frac{1}{a^m}$

Example 5a (Easy): Simplify

5^{-2}

= \frac{1}{25}

Example 5b (Difficult): Simplify

(\frac{2x^{-3}}{y^2})^{-2}

= \frac{y^4}{4x^6}

Power of a Product Rule

$(ab)^m = a^m \times b^m$

Example 6a (Easy): Simplify

(3x)^2

= 3^2 \times x^2 = 9x^2

Example 6b (Difficult): Expand

(2a^2 b)^3

= 8a^6 b^3

Power of a Quotient Rule

$\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$

Example 7a (Easy): Simplify

(\frac{4}{5})^2

= \frac{16}{25}

Example 7b (Difficult): Simplify

(\frac{x^2}{2y})^3

= \frac{x^6}{8y^3}

Combining Multiple Laws

Example 8a (Easy): Simplify

(2^3 \cdot 2^{-1})^2

= (2^2)^2 = 2^4 = 16

Example 8b (Difficult): Simplify

(\frac{a^3 b^{-2}}{c^{-1}})^2

= \frac{a^6 c^2}{b^4}

Example 9a (Easy): Simplify

x^6 \div x^2

= x^4

Example 9b (Difficult): Simplify

(\frac{x^3 y^2}{z})^3

= \frac{x^9 y^6}{z^3}

Example 10a (Easy): Evaluate

(2x)^2 \cdot (3x)^2

= 4x^2 \cdot 9x^2 = 36x^4

Example 10b (Difficult): Simplify

\frac{(2x^{-1})^2}{(x^2 y^{-3})^0}

= \frac{4x^{-2}}{1} = \frac{4}{x^2}

Exercises

Simplify $4^3 \times 4^2$ .
Simplify $\frac{6^5}{6^2}$ .
[WAEC] Simplify $(x^3)^4$ . (Past Question)
[NECO] Evaluate $7^0 + 5^0$ . (Past Question)
[JAMB] Simplify $3^{-2} \cdot 3^5$ . (Past Question)
Evaluate $(2a^2b)^3$ .
Simplify $(\frac{5}{2})^2$ .
Simplify $(10^{-1} \cdot 10^3)^2$ .
[WAEC] Simplify $\frac{x^6}{x^2} \div x$ . (Past Question)
[NABTEB] Evaluate $(\frac{2x^2}{y})^3$ . (Past Question)

Conclusion / Recap

The laws of indices help us simplify and evaluate expressions involving powers. These include the product, quotient, zero, negative, and power rules. Mastery of these rules supports further study in algebra, logarithms, and exponential functions.