Indices | Skymenu Mathematics
LESSON 6 • NUMBER

Indices

Learn how indices, or powers, help us write repeated multiplication in a shorter and more useful form.

📘 Number ⏱ Beginner friendly ✏️ Practice included
KEY IDEA an

Base a, raised to the power n.

What you will learn

01Understand the meaning of indices.
02Identify bases and powers.
03Use the laws of indices.
04Work with zero and negative indices.
05Simplify expressions involving powers.
06Apply indices to everyday mathematics.
01

What Are Indices?

An index tells us how many times a number is multiplied by itself. Indices are also called powers or exponents.

Example: 24 means 2 × 2 × 2 × 2 = 16.
33 = 3 × 3 × 3 = 27
02

Base and Index

In an expression such as 53, the number 5 is the base and 3 is the index.

Remember

53 means 5 is used as a factor three times.

5 × 5 × 5 = 125

03

Multiplying Powers with the Same Base

When multiplying powers with the same base, add the indices.

Law: am × an = am+n

Example

23 × 24 = 23+4 = 27 = 128

04

Dividing Powers with the Same Base

When dividing powers with the same base, subtract the indices.

Law: am ÷ an = am−n, for a ≠ 0.

Example

56 ÷ 52 = 56−2 = 54 = 625

05

A Power Raised to Another Power

When a power is raised to another power, multiply the indices.

Law: (am)n = amn

Example

(23)2 = 23×2 = 26 = 64

06

Zero Index

Any non-zero number raised to the power zero equals 1.

a0 = 1   (a ≠ 0)

70 = 1

1000 = 1

07

Negative Indices

A negative index means we use the reciprocal of the corresponding positive power.

Law: a−n = 1/an, for a ≠ 0.

Example

2−3 = 1/23 = 1/8

08

Power of a Product

When a product is raised to a power, the power applies to each factor.

Law: (ab)n = anbn

Example

(2 × 3)2 = 22 × 32 = 4 × 9 = 36

09

Simplifying Expressions with Indices

Use the correct index law and then simplify the numerical or algebraic expression.

Example 1

x3 × x2 = x5

Example 2

y8 ÷ y3 = y5

Example 3

(m2)4 = m8

10

Common Mistakes

❌ Multiplying instead of adding

a2 × a3 is a5, not a6.

❌ Adding during division

a6 ÷ a2 is a4, not a8.

❌ Forgetting brackets

(a2)3 gives a6.

❌ Treating a0 as zero

For a ≠ 0, a0 = 1.

11

Quick Practice

Try these questions before checking the answers.

  1. Write 4 × 4 × 4 × 4 as an index.
  2. Calculate 34.
  3. Simplify: 23 × 25.
  4. Simplify: x9 ÷ x4.
  5. Simplify: (a3)2.
  6. Calculate 80.
  7. Write 5−2 as a fraction.
  8. Simplify: (2x)3.
Show answers
  1. 44
  2. 81
  3. 28 = 256
  4. x5
  5. a6
  6. 1
  7. 1/25
  8. 8x3
12

Lesson Summary

Multiplication: am × an = am+n

Division: am ÷ an = am−n

Power of a power: (am)n = amn

Zero index: a0 = 1

Negative index: a−n = 1/an

Power of a product: (ab)n = anbn

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