Laws of Indices

Master the fundamental rules for working with powers and exponents for all rational values

SPEC 2.1 • ALGEBRA AND FUNCTIONS

What Are Indices?

Definition

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

2³ = 2 × 2 × 2 = 8

2 is the base | 3 is the index (power)

Examples

ExpressionMeaningValue5²5 × 5254³4 × 4 × 46410⁴10 × 10 × 10 × 1010000

The Three Basic Index Laws

Law 1: Multiplication Law

aᵐ × aⁿ = aᵐ⁺ⁿ

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

Law 2: Division Law

aᵐ ÷ aⁿ = aᵐ⁻ⁿ

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

Law 3: Power Law

(aᵐ)ⁿ = aᵐⁿ

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

Important

These laws only work when the bases are the same. You cannot simplify 2³ × 3⁴ using these laws.

Zero and Negative Indices

Zero Index

a⁰ = 1 (where a ≠ 0)Why? Using the division law: aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰, but also aⁿ ÷ aⁿ = 1

Negative Index

a⁻ⁿ = 1/aⁿWhy? Using the division law: a⁰ ÷ aⁿ = a⁰⁻ⁿ = a⁻ⁿ, but also 1/aⁿ

Tip

A negative index means "take the reciprocal". The negative sign does NOT make the answer negative!

Example: 2⁻³ = 1/2³ = 1/8 (positive answer!)

Fractional Indices

Unit Fraction Index = Roots

a^½ = √asquare roota^⅓ = ³√acube roota^¼ = ⁴√afourth root

General Fractional Index

a^m/n = (ⁿ√a)ᵐ = ⁿ√(aᵐ) The denominator is the root, the numerator is the power

Strategy - Root First!

It's usually easier to take the root first, then raise to the power:

8^⅔ = (³√8)² = 2² = 4 ✓

Negative Fractional Indices

Combining Negative and Fractional

a^−m/n = 1 ÷ a^m/n = 1 ÷ (ⁿ√a)ᵐ

Step-by-Step Method

Ignore the negative: Work out the positive power first

Take the root: Find the nth root of the base

Apply the power: Raise to the power m

Flip it: Take the reciprocal (because of the negative)

Example

16^−¾ → 16^¾ first → (⁴√16)³ = 2³ = 8 → flip → ⅛

01Worked Example

Simplify the following:

2x³ × 4x⁵

1

Multiply the coefficients: 2 × 4 = 8

2

Apply the multiplication law to the x terms: x³ × x⁵ = x³⁺⁵ = x⁸

3

Combine the results

Answer: 8x⁸

02Worked Example

Evaluate the following:

5⁰ + 3⁻²

1

Evaluate 5⁰: Any non-zero number to the power 0 equals 1, so 5⁰ = 1

2

Evaluate 3⁻²: Using a⁻ⁿ = 1/aⁿ, we get 3⁻² = 1/3² = 1/9

3

Add the results: 1 + 1/9 = 9/9 + 1/9 = 10/9

Answer: 10/9

03Worked Example

Evaluate without a calculator:

27^⅔

1

The denominator 3 means cube root, the numerator 2 means square

2

Find the cube root of 27 first: ³√27 = 3 (since 3 × 3 × 3 = 27)

3

Square the result: 3² = 9

Answer: 9

04Worked Example

Evaluate without a calculator:

16^−¾

1

Work out 16^¾ first (ignore the negative for now)

2

The 4 means fourth root: ⁴√16 = 2 (since 2⁴ = 16)

3

Cube it (the 3): 2³ = 8

4

Now flip it (because of the negative): 1/8

Answer: ⅛

Simplifying Algebraic Expressions

When simplifying algebraic expressions with indices, remember:

Key Rules

Apply index laws to each variable separately

Deal with coefficients using normal arithmetic

(ab)ⁿ = aⁿbⁿ — power applies to everything inside the bracket

(a/b)ⁿ = aⁿ/bⁿ — same rule for fractions

Strategy

For complex expressions, work systematically:

Deal with brackets first (using the power law)

Convert any roots to fractional indices

Apply multiplication/division laws

Write your final answer with positive indices

05Worked Example

Simplify fully:

12x⁵y⁻² ÷ 4x⁻¹y³

1

Divide the coefficients: 12 ÷ 4 = 3

2

Apply division law to x: x⁵ ÷ x⁻¹ = x⁵⁻⁽⁻¹⁾ = x⁵⁺¹ = x⁶

3

Apply division law to y: y⁻² ÷ y³ = y⁻²⁻³ = y⁻⁵

4

Write with positive indices: y⁻⁵ = 1/y⁵

Answer: 3x⁶/y⁵

06Worked Example

Express in the form axⁿ:

5√x

1

Recall that √x = x^½ (square root = power of one half)

2

Substitute: 5√x = 5 × x^½

3

Write in required form

Answer: 5x^½

07Worked Example

Express in the form axⁿ:

4/³√x

1

Recall that ³√x = x^⅓ (cube root = power of one third)

2

Rewrite: 4/³√x = 4/x^⅓ = 4 × x^−⅓

3

Dividing by x^⅓ is the same as multiplying by x^−⅓

Answer: 4x^−⅓

08Worked Example

Simplify:

√x × x³

1

Convert √x to index form: √x = x^½

2

Apply multiplication law: x^½ × x³ = x^½ + 3

3

Add the indices: ½ + 3 = ½ + 6/2 = 7/2

Answer: x^7/2 (or equivalently x³√x)

09Worked Example

Simplify:

x⁴ ÷ √x

1

Convert √x to index form: √x = x^½

2

Apply division law: x⁴ ÷ x^½ = x^4 − ½

3

Subtract the indices: 4 − ½ = 8/2 − ½ = 7/2

Answer: x^7/2 (or equivalently x³√x)

10

Worked Example

Simplify fully, giving your answer with positive indices:

(2x³)² × √x ÷ 4x⁵

1

Expand (2x³)²: = 2² × (x³)² = 4x⁶

2

Convert √x = x^½

3

Numerator: 4x⁶ × x^½ = 4x^6+½ = 4x^13/2

4

Divide: 4x^13/2 ÷ 4x⁵ = x^13/2 − 5 = x^13/2 − 10/2 = x^3/2

Answer: x^3/2 (or equivalently x√x)

Solving Equations with Indices

There are two main types of equations involving indices:

Type 1: Unknown in the Index

Make the bases the same, then equate indices.

Example: 2ˣ = 8 becomes 2ˣ = 2³, so x = 3

Type 2: Unknown in the Base

Use the relationship between indices and roots.

Example: x³ = 27 gives x = ³√27 = 3

Key Powers to Know

2: 2, 4, 8, 16, 32, 64, 128 | 3: 3, 9, 27, 81 | 5: 5, 25, 125

11

Worked Example

Solve the equation:

4²ˣ⁺¹ = 8ˣ

1

Express both sides with base 2: 4 = 2² and 8 = 2³

2

Rewrite: (2²)²ˣ⁺¹ = (2³)ˣ

3

Apply power law: 2²⁽²ˣ⁺¹⁾ = 2³ˣ

4

Simplify left side: 2⁴ˣ⁺² = 2³ˣ

5

Since bases are equal, indices must be equal: 4x + 2 = 3x

6

Solve: 4x − 3x = −2, so x = −2

Answer: x = −2

12

Worked Example

Solve the equation:

x^3/2 = 64

1

To isolate x, raise both sides to the reciprocal power ⅔

2

(x^3/2)^⅔ = 64^⅔

3

Left side: x^(3/2 × ⅔) = x¹ = x

4

Right side: 64^⅔ = (³√64)² = 4² = 16

Answer: x = 16

Practice

Question 1: Simplify fully: 6a⁴b⁻³ ÷ 2a⁻²b²

Step 1: Divide coefficients: 6 ÷ 2 = 3

Step 2: Apply division law to a: a⁴ ÷ a⁻² = a⁴⁻⁽⁻²⁾ = a⁶

Step 3: Apply division law to b: b⁻³ ÷ b² = b⁻³⁻² = b⁻⁵

Step 4: Write with positive indices: 3a⁶/b⁵

Answer: 3a⁶/b⁵

Practice

Question 2: Evaluate without a calculator: 125^−⅔

Step 1: Work out 125^⅔ first (ignore the negative)

Step 2: The 3 means cube root: ³√125 = 5 (since 5³ = 125)

Step 3: Square it (the 2): 5² = 25

Step 4: Flip it (because of the negative): 1/25

Answer: 1/25

Practice

Question 3: Solve the equation: 9ˣ⁻¹ = 27ˣ

Step 1: Express with base 3: 9 = 3² and 27 = 3³

Step 2: Rewrite: (3²)ˣ⁻¹ = (3³)ˣ

Step 3: Apply power law: 3²⁽ˣ⁻¹⁾ = 3³ˣ

Step 4: Simplify: 3²ˣ⁻² = 3³ˣ

Step 5: Equate indices: 2x − 2 = 3x

Step 6: Solve: −2 = 3x − 2x, so x = −2

Answer: x = −2

Summary

The Index Laws

LawFormulaRemember

Multiplicationaᵐ × aⁿ = aᵐ⁺ⁿAdd indices

Divisionaᵐ ÷ aⁿ = aᵐ⁻ⁿSubtract indices

Power(aᵐ)ⁿ = aᵐⁿMultiply indices

Zero indexa⁰ = 1Always 1 (if a ≠ 0)

Negative indexa⁻ⁿ = 1/aⁿFlip it

Fractional indexa^m/n = (ⁿ√a)ᵐRoot then power

Key Exam Skills

Convert roots to fractional indices: √x = x^½, ³√x = x^⅓

Simplify expressions by applying index laws systematically

Express answers with positive indices unless stated otherwise

Solve equations by making bases equal or using reciprocal powers