Index laws from repeated factors
Explain positive, zero, negative and fractional powers using products and inverse operations.
Core: positive, zero and negative integer indices. Extended also includes fractional indices.
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
From guided practice to a new situation
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.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
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.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs contains 21; Coordinate geometry contains 10 teaching lessons and a mixed checkpoint. Nine earlier overviews support selected topics across the syllabus. The other six syllabus areas, full cumulative assessment and human teacher review are not yet complete. Written lesson practice is self-checked, not automatically graded. The checkpoint samples skills and does not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources