Index laws and roots | OCR GCSE Maths Notes

Visual model

Index laws count repeated multiplication

a^{3} \times a^{2}

a^{5}

same base: add the powers

Gold-standard guide

20 mins

What you will learn

Calculate with powers, roots and index rules.

Use a clear step-by-step method for index laws and roots.

Check your answer and avoid the most common exam mistake.

Useful before you start

Core number skillsEarlier number skillsShowing clear working

Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Index laws: multiply → add powers; divide → subtract powers; power of a power → multiply powers

Step 1

Apply the power of a bracket — multiply each index inside by 2

(3x²y³)² = 3² × x^(2×2) × y^(3×2) = 9x⁴y⁶

Step 2

Multiply by 2x⁻¹y — add indices of matching bases

9x⁴y⁶ × 2x⁻¹y = 18 × x^(4+(−1)) × y^(6+1) = 18x³y⁷

Step 3

Check for negative or zero indices in the final answer

All indices are positive, so no further simplification needed

Watch out

Students forget that the coefficient (the number in front) must also be raised to the power when squaring a bracket

Multiply powers

$a^m \times a^n = a^(m+n).$

Divide powers

$a^\frac{m}{a}^n = a^(m-n).$

Power of a power

$(a^m)^n = a^(mn).$

Worked example

Simplify (3x²y³)² × 2x⁻¹y

Apply the power of a bracket — multiply each index inside by 2: (3x²y³)² = 3² × x^(2×2) × y^(3×2) = 9x⁴y⁶.We multiply every index inside the bracket by the outside power because (ab)ⁿ = aⁿbⁿ.

Multiply by 2x⁻¹y — add indices of matching bases: 9x⁴y⁶ × 2x⁻¹y = 18 × x^(4+(−1)) × y^(6+1) = 18x³y⁷.We add powers because we are multiplying the same base.

Check for negative or zero indices in the final answer: All indices are positive, so no further simplification needed.

Final answer

18x³y⁷

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

Simplify (3x²y³)² × 2x⁻¹y

3 marks4 minsindex-laws-and-roots-worked

Show solution

Worked solution

1.Apply the power of a bracket — multiply each index inside by 2: (3x²y³)² = 3² × x^(2×2) × y^(3×2) = 9x⁴y⁶.We multiply every index inside the bracket by the outside power because (ab)ⁿ = aⁿbⁿ.
2.Multiply by 2x⁻¹y — add indices of matching bases: 9x⁴y⁶ × 2x⁻¹y = 18 × x^(4+(−1)) × y^(6+1) = 18x³y⁷.We add powers because we are multiplying the same base.
3.Check for negative or zero indices in the final answer: All indices are positive, so no further simplification needed.

Final answer

18x³y⁷

Mark points

M1: apply the power of a bracket — multiply each index inside by 2
M1: multiply by 2x⁻¹y — add indices of matching bases
M1: check for negative or zero indices in the final answer
A1: 18x³y⁷

Watch out

Students forget that the coefficient (the number in front) must also be raised to the power when squaring a bracket.(3x²)² = 9x⁴, not 3x⁴. The index applies to every factor inside the bracket, including the number.

Diagnosticrecall

Simplify 5⁴ ÷ 5²

1 mark2 minsindex-laws-and-roots-q1

Show solution

Worked solution

1.Spot the skill: Index laws: multiply → add powers; divide → subtract powers; power of a power → multiply powers.
2.Use the apply the power of a bracket — multiply each index inside by 2 stage first, then multiply by 2x⁻¹y — add indices of matching bases.
3.Keep the final answer visible: 5² = 25.

Final answer

5² = 25

Mark points

M1: use the correct index laws: multiply → add powers; divide → subtract powers; power of a power → multiply powers.these rules only apply when the base is the same.think of the index as a count of how many times the base is multiplied by itself.
A1: 5² = 25

Watch out

Easyprocedure

Simplify (2³)⁴

2 marks3 minsindex-laws-and-roots-q2

Show solution

Worked solution

1.Spot the skill: Index laws: multiply → add powers; divide → subtract powers; power of a power → multiply powers.
2.Use the multiply by 2x⁻¹y — add indices of matching bases stage first, then check for negative or zero indices in the final answer.
3.Keep the final answer visible: 2¹² = 4096.

Final answer

2¹² = 4096

Mark points

M1: use the correct index laws: multiply → add powers; divide → subtract powers; power of a power → multiply powers.these rules only apply when the base is the same.think of the index as a count of how many times the base is multiplied by itself.
A1: 2¹² = 4096

Watch out

Mediumreasoning

Evaluate 27^( $\frac{1}{3}$ )

3 marks4 minsindex-laws-and-roots-q3

Show solution

Worked solution

1.Spot the skill: Index laws: multiply → add powers; divide → subtract powers; power of a power → multiply powers.
2.Use the check for negative or zero indices in the final answer stage first, then apply the power of a bracket — multiply each index inside by 2.
3.Keep the final answer visible: 3.

Final answer

Mark points

M1: use the correct index laws: multiply → add powers; divide → subtract powers; power of a power → multiply powers.these rules only apply when the base is the same.think of the index as a count of how many times the base is multiplied by itself.
A1: 3

Watch out

Hardproblem solving

Simplify 4x³y² × 3x²y⁻¹

3 marks5 minsindex-laws-and-roots-q4

Show solution

Worked solution

1.Spot the skill: Index laws: multiply → add powers; divide → subtract powers; power of a power → multiply powers.
2.Use the apply the power of a bracket — multiply each index inside by 2 stage first, then multiply by 2x⁻¹y — add indices of matching bases.
3.Keep the final answer visible: 12x⁵y.

Final answer

12x⁵y

Mark points

M1: use the correct index laws: multiply → add powers; divide → subtract powers; power of a power → multiply powers.these rules only apply when the base is the same.think of the index as a count of how many times the base is multiplied by itself.
A1: 12x⁵y

Watch out

Exam-stylemulti-step

Evaluate 16^( $\frac{3}{4}$ )

4 marks6 minsindex-laws-and-roots-q5

Show solution

Worked solution

1.Spot the skill: Index laws: multiply → add powers; divide → subtract powers; power of a power → multiply powers.
2.Use the multiply by 2x⁻¹y — add indices of matching bases stage first, then check for negative or zero indices in the final answer.
3.Keep the final answer visible: 8.

Final answer

Mark points

M1: use the correct index laws: multiply → add powers; divide → subtract powers; power of a power → multiply powers.these rules only apply when the base is the same.think of the index as a count of how many times the base is multiplied by itself.
A1: 8

Watch out

Grade 9 stretchproblem solving

Solve 3^(x + 1) = 81.

4 marks7 minsindices-g9

Show solution

Worked solution

1.Write 81 as a power of 3.
2.Equate the powers.

Final answer

x = 3

Mark points

M1: 81 = 3⁴
A1: x + 1 = 4, so x = 3

Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Hard exam-stylemulti-step problem

Simplify (3⁵ × 3^-2) / 3⁴. Give your answer as a fraction.

3 marks6 minsindices-paper

Show solution

Worked solution

1.Add powers when multiplying.
2.Subtract the denominator power.
3.Rewrite the negative power as a fraction.

Final answer

$\frac{1}{3}$

Mark points

M1: use 3^(5 - 2 - 4)
M1: obtain 3^-1
A1: obtain $\frac{1}{3}$

Watch out

Read the full question before calculating. Keep each stage of your working visible.

Timed checkpoint

12 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Index laws and roots - 2 marksSimplify 5⁴ ÷ 5²Mark answer

Answer

5² = 25

2Calculations and order of operations - 2 marksWork out (6 + 2) × 3 − 5Mark answer

Answer

3Integers, decimals and place value - 2 marksOrder these from smallest to largest: 0.3, 0.03, 0.303, 0.033Mark answer

Answer

0.03, 0.033, 0.3, 0.303

4Fractions - 3 marksA recipe needs ¾ cup of sugar. How much sugar is needed for 2½ batches?Mark answer

Answer

1⅞ cups

Mastery check

I can explain the method for index laws and roots.
I can show clear working without skipping key steps.
I can avoid this mistake: Students forget that the coefficient (the number in front) must also be raised to the power when squaring a bracket.(3x²)² = 9x⁴, not 3x⁴. The index applies to every factor inside the bracket, including the number.

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