GCSE · Maths · AQA · Spec 8300 · Higher

Fractional indices (Higher)

Half of 16 is 8. So 16 to the power of a half must be 8… right? Put your guess on the line and find out.

Guess first

Where do the powers of 16 land?

You already know 16⁰ = 1 and 16¹ = 16. Now the power is a fraction. Before any rules, where do you think 16¹⁄⁴, 16¹⁄² and 16³⁄⁴ sit? Drag all five onto the line, then check.

Maths · Algebra

Why ½ has to mean square root

Use a law you already trust, adding the powers when you multiply, and watch it decide the answer for you.

GoalWhat number is a¹⁄²?
1
a¹⁄² × a¹⁄²

Whatever a¹⁄² is, multiply it by itself.

2
3
4

Step 1 of 4

Whatever a¹⁄² is, multiply it by itself.

Predict, then check

Now the top of the fraction isn't 1. In 8²⁄³ the 3 on the bottom means cube root and the 2 on top means square. But which comes first?

Amir works out 8²⁄³ as (∛8)²: cube root first, then square. Beth works it out as ∛(8²): square first, then cube root. What happens?

Watch the method

Problem

Work out (a) 27²⁄³ and (b) 25⁻³⁄² without a calculator.

Your turn

Fill in the missing steps

Work out 32⁻²⁄⁵ without a calculator.

  1. 32⁻²⁄⁵Three instructions: a minus sign, a 5 on the bottom and a 2 on top.
  2. missing step
Which line is step 2?

Spot the mistake

Where does this answer go wrong?

Simplify (x¹⁄² × x¹⁄³)⁶, giving your answer as a power of x.

A student's answer — which line goes wrong?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

aᵐ⁄ⁿ = (ⁿ√a)ᵐ

The bottom of the fraction is the root. The top is the power. Take the root first and the numbers stay small.

What you need to know

  • What a fraction in the index means: the denominator is a root and the numerator is a power.
  • How to work out a fractional power without a calculator, including one with a negative index.
  • Why a¹⁄ⁿ has to be the nth root: it's the only meaning that keeps the index laws true.
  • How to use the index laws when the indices are fractions.

The big picture

A fraction in the index is two instructions, not a multiplier. The denominator is a root and the numerator is a power, so aᵐ⁄ⁿ = (ⁿ√a)ᵐ. That makes 16¹⁄² = 4, not 8, and 27²⁄³ = (∛27)² = 9. These meanings aren't arbitrary: a¹⁄² × a¹⁄² = a¹ = a, so a¹⁄² has to be √a. A negative index still means 'one over', and every index law works with fractional indices, as long as the fraction arithmetic inside them is right.

Key points

1a¹⁄² = √a and a¹⁄³ = ∛a. In general, a¹⁄ⁿ = ⁿ√a.
2aᵐ⁄ⁿ = (ⁿ√a)ᵐ: the denominator n is the root and the numerator m is the power.
3A fractional index is not a multiplier: 25¹⁄² = √25 = 5, not 25 × ½ = 12.5.
4Take the root first to keep the numbers small. For a positive base, the order doesn't change the answer.
5A negative index means 'one over': a⁻ᵐ⁄ⁿ = 1 ÷ aᵐ⁄ⁿ. It doesn't make the answer negative.
6The index laws still apply: multiplying adds the indices, dividing subtracts them, a power of a power multiplies them. Add and subtract fractional indices with a common denominator.
7a¹⁄² × a¹⁄² = a¹ = a, which is why a¹⁄² must be √a. Zero, negative and fractional indices all mean whatever keeps the index laws true.

Worked example

Problem

Write 4√8 as a single power of 2.

⚠ Watch out

Treating the index as a multiplier. 25¹⁄² is not 25 × ½ = 12.5. The ½ means square root, so 25¹⁄² = 5. If your answer is just the base times the fraction, stop and read the index again.

🧠

Memory hook

Think of a tree: its roots are at the bottom. The bottom of the fraction is the root; the top is the power.

✓

Check yourself

Work out 1000²⁄³ without a calculator. (Check: ∛1000 = 10, then 10² = 100.)

Flashcards

(13)
Is 25¹⁄² equal to 12.5?
No. 25¹⁄² = √25 = 5. A fractional index is not something you multiply the base by.
What does a¹⁄ⁿ mean?
The nth root of a. So a¹⁄² = √a and a¹⁄³ = ∛a.
In aᵐ⁄ⁿ, which part is the root and which is the power?
The denominator n is the root; the numerator m is the power: aᵐ⁄ⁿ = (ⁿ√a)ᵐ.
Why must a¹⁄² be √a?
a¹⁄² × a¹⁄² = a¹ = a, so a¹⁄² is the number that multiplies by itself to make a.
Root first or power first?
For a positive base both give the same answer, but root first keeps the numbers small.
What does a minus sign in the index do?
It means 'one over': a⁻ᵐ⁄ⁿ = 1 ÷ aᵐ⁄ⁿ. It does not mean the answer is negative.
Work out 100³⁄².
√100 = 10, then 10³ = 1000.
How do you multiply x¹⁄² × x¹⁄³?
Add the indices with a common denominator: 3/6 + 2/6 = 5/6, so x⁵⁄⁶.
Simplify x³⁄⁴ ÷ x¹⁄⁴.
Subtract the indices: 3/4 − 1/4 = 2/4 = 1/2, so x¹⁄².
Simplify (x²⁄³)³.
Power of a power: multiply the indices. 2/3 × 3 = 2, so x².
Write ∛25 as a power of 5.
25 = 5², so ∛25 = (5²)¹⁄³ = 5²⁄³.
Why is a⁰ = 1 (for a ≠ 0)?
aᵐ ÷ aᵐ = aᵐ⁻ᵐ = a⁰, and anything non-zero divided by itself is 1. Like fractional indices, it's the meaning that keeps the laws true.
What happens to 16ˣ as x goes 0, ¼, ½, ¾, 1?
1, 2, 4, 8, 16. Each equal step in the index doubles the value here: it multiplies, it doesn't add.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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