Indices and Roots

This lesson covers the laws of indices (also called powers or exponents) and roots for AQA GCSE Mathematics. Understanding index laws is essential for simplifying expressions, working with standard form, and tackling algebraic problems. Some rules — particularly negative and fractional indices — are higher-tier content.

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What Are Indices?

An index (plural: indices) tells you how many times to multiply a number by itself.

• 2 cubed = 2 x 2 x 2 = 8
• 5 squared = 5 x 5 = 25
• 10 to the power of 4 = 10 x 10 x 10 x 10 = 10,000

Expression	Meaning	Value
3 squared	3 x 3	9
2 to the power of 5	2 x 2 x 2 x 2 x 2	32
10 cubed	10 x 10 x 10	1,000
7 to the power of 1	7	7
4 to the power of 0	(see below)	1

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The Laws of Indices

These laws apply when the base number is the same.

Law 1: Multiplying — Add the Powers

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

a to the power m x a to the power n = a to the power (m + n)

Example: 3 to the power 4 x 3 squared = 3 to the power (4 + 2) = 3 to the power 6

Law 2: Dividing — Subtract the Powers

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

a to the power m divided by a to the power n = a to the power (m - n)

Example: 5 to the power 7 / 5 cubed = 5 to the power (7 - 3) = 5 to the power 4

Law 3: Power of a Power — Multiply the Powers

When raising a power to another power, multiply the indices.

(a to the power m) to the power n = a to the power (m x n)

Example: (2 cubed) to the power 4 = 2 to the power (3 x 4) = 2 to the power 12

graph TD
    A["Index Laws Summary"]
    A --> B["Multiply: ADD powers"]
    A --> C["Divide: SUBTRACT powers"]
    A --> D["Power of power: MULTIPLY powers"]
    A --> E["Power of 0: always equals 1"]
    A --> F["Negative power: reciprocal"]
    A --> G["Fractional power: root"]

Exam Tip: The index laws only work when the base numbers are the same. You cannot use them to simplify expressions like 2 cubed x 3 squared.

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The Zero Index

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

a to the power 0 = 1 (where a is not zero)

Why? Using the division law: a to the power n / a to the power n = a to the power (n - n) = a to the power 0. But any number divided by itself is 1. Therefore a to the power 0 = 1.

Examples:

• 5 to the power 0 = 1
• 100 to the power 0 = 1
• (-3) to the power 0 = 1

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Negative Indices [H]

A negative index means one over (the reciprocal of) the positive power.

a to the power (-n) = 1 / (a to the power n)

Worked Examples

Expression	Working	Value
2 to the power (-3)	1 / (2 cubed)	1/8
5 to the power (-2)	1 / (5 squared)	1/25
10 to the power (-1)	1 / 10	0.1
(1/3) to the power (-2)	(3/1) squared = 9	9

Exam Tip: A negative index does NOT make the answer negative. It creates a reciprocal. This is a very common misconception.

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Fractional Indices [H]

A fractional index means a root.

a to the power (1/n) = the nth root of a

• a to the power (1/2) = the square root of a
• a to the power (1/3) = the cube root of a
• a to the power (1/4) = the fourth root of a

Combining Fractional and Integer Indices

a to the power (m/n) = the nth root of (a to the power m)

Or equivalently: (the nth root of a) to the power m

Worked Examples

Expression	Working	Value
25 to the power (1/2)	square root of 25	5
8 to the power (1/3)	cube root of 8	2
16 to the power (3/4)	(4th root of 16) cubed = 2 cubed	8
27 to the power (2/3)	(cube root of 27) squared = 3 squared	9
32 to the power (3/5)	(5th root of 32) cubed = 2 cubed	8

Step-by-Step for Harder Examples

Evaluate 125 to the power (-2/3).

Step 1: Deal with the negative sign — take the reciprocal: 1 / (125 to the power (2/3))

Step 2: Deal with the fraction — cube root first, then square:

• Cube root of 125 = 5
• 5 squared = 25

Step 3: Apply the reciprocal: 1/25

Answer: 1/25

Exam Tip: For fractional indices, always do the ROOT first, then the POWER. This keeps numbers small and manageable. Then apply any negative sign as a reciprocal at the end.

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Square Roots and Cube Roots

Square Roots

The square root of a number n is the value that, when multiplied by itself, gives n.

• Square root of 49 = 7 (because 7 x 7 = 49)
• Square root of 144 = 12 (because 12 x 12 = 144)

Perfect squares to know: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225

Cube Roots