Index laws from repeated factors
Explain positive, zero, negative and fractional powers using products and inverse operations.
Section Paper 3 · Core · Calculator
Core: positive, zero and negative integer indices. Extended also includes fractional indices. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Read a positive power as repeated multiplication.
- Find square roots, cube roots and reciprocals.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is 2³ × 2²?
What you will learn
- Use the index laws for a common base.
- Interpret zero and negative integer powers.
- Use fractional powers in Extended questions.
A common base is essential
Multiplying 2³ by 2⁴ joins three factors of two to four more, giving 2⁷. Thus aᵐ × aⁿ = aᵐ⁺ⁿ when the powers are defined. Dividing cancels factors: 2⁵ ÷ 2² = 2³. The common base matters; 2³ × 3⁴ cannot be combined by simply adding exponents.
Raising a power to a power repeats the whole product. For (3²)⁴ there are four groups of two factors of three, giving 3⁸. Hence (aᵐ)ⁿ = aᵐⁿ for the integer powers used here. A product also distributes through an integer power: (2 × 5)³ = 2³ × 5³.
Zero and negative powers preserve the pattern
For a non-zero base, a³ ÷ a³ equals one. The quotient law gives a³⁻³ = a⁰, so a⁰ = 1. This does not say that zero to every power is one; the non-zero condition matters. Zero to a negative power is undefined because it would require division by zero.
Reducing a power by one divides by the base. For base two the sequence 2², 2¹, 2⁰, 2⁻¹, 2⁻² is 4, 2, 1, 1/2, 1/4. Therefore a⁻ⁿ = 1/aⁿ for non-zero a. A negative exponent describes a reciprocal, not a negative answer.
Extended: roots are fractional powers
For positive a, a¹⁄² is √a and a¹⁄³ is ∛a. More generally, aᵐ⁄ⁿ means take the positive nth root and raise it to m. Thus 64²⁄³ = (∛64)² = 4² = 16. Using a positive base avoids extra real-number domain restrictions in these examples.
A negative fractional exponent adds a reciprocal: 16⁻³⁄⁴ = 1/(16³⁄⁴) = 1/8. Find the fourth root first: 16¹⁄⁴ = 2, then cube it. The denominator of the exponent determines the root; the numerator determines the power. Do not multiply the base by the fraction.
Keep brackets and exact values
The base of (−3)² is negative three, so its square is nine. In −3² the power acts on three before the minus sign, giving negative nine. When dividing fractions raised to a negative power, invert the complete fraction: (2/3)⁻² = (3/2)² = 9/4.
Index laws simplify exact expressions before a decimal approximation is needed. For 5⁶/5⁴, cancel to 5² rather than evaluating two large numbers. Check a law with a small numerical example whenever you are unsure; it can reveal whether you have added, subtracted or multiplied exponents incorrectly.
Pause and explain
What is 5⁻²?
Worked example
Calculate 3⁴ × 3⁻² ÷ 3.
Show the worked solution
- Write the final divisor as 3¹.
- Combine the exponents: 4 + (−2) − 1 = 1.
- The expression is 3¹ = 3. Check with 81 × (1/9) ÷ 3.
Answer 3
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Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Simplify (2³)² ÷ 2⁴.
Give me a hint
Multiply the nested exponents, then subtract the divisor's exponent.
Compare my reasoning
- (2³)² = 2⁶.
- 2⁶ ÷ 2⁴ = 2².
- 2² = 4; the original quotient is 64/16, which agrees.
4
Look for these in your work
- Multiplied exponents for the nested power.
- Subtracted exponents for division with a common base.
Find 7⁰, 2⁻³ and (2/3)⁻² exactly.
Give me a hint
A negative power takes the reciprocal of the whole base.
Compare my reasoning
- 7⁰ = 1 because the base is non-zero.
- 2⁻³ = 1/2³ = 1/8.
- (2/3)⁻² = (3/2)² = 9/4.
1; 1/8; 9/4
Look for these in your work
- Did not turn a negative exponent into a negative value.
- Inverted both parts of the fraction.
Extended: a cube has volume 64 cm³. Find 64²⁄³ and explain which geometric quantity it represents.
Give me a hint
Take the cube root before squaring; distinguish one face from the whole surface.
Compare my reasoning
- The side is ∛64 = 4 cm.
- 64²⁄³ = 4² = 16.
- This is the area of one face, 16 cm². Total surface area would be six times that.
16; one face has area 16 cm².
Look for these in your work
- Linked the fractional power to root then square.
- Interpreted the result as area, not volume or total surface area.
Common mistakes
- Adding exponents when the bases differ.
- Treating a negative exponent as a negative sign.
- Using a⁰ = 1 when a is zero.
What can you explain now?
Extended: calculate 81³⁄⁴.
Compare with the explanation
27
The fourth root of 81 is 3; cubing it gives 27.
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Make it stick
Explain the integer index laws using repeated factors. Extended learners: evaluate 32²⁄⁵ without a calculator and explain both numbers in the exponent.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.