GCSE · Maths · AQA · Spec 8300 · Foundation

Laws of indices in algebraic expressions

Quick: what's a² × a³? If you thought a⁶, write it out as a·a × a·a·a and count. Five a's, so a⁵. Every law of indices is just that: counting.

Count the factors

Which single power does each one make?

Pick an expression, then put it in the bin for the power it simplifies to. Commit first: the reason appears once you've placed it.

Still to sort

One power, index 5 (0)

Ends up as a single power such as a⁵, b⁵ or 3⁵.

Where the line is: Multiplying same-base powers ADDS the counts: a² × a³ is 2 + 3 = 5 factors. You don't multiply the indices here.

One power, index 6 (0)

Ends up as a single power such as a⁶ or 3⁶.

Where the line is: (a²)³ means three copies of a² multiplied together: 2 + 2 + 2 = 6 factors. Adding the same number again and again is multiplying, so here the indices multiply.

Can't become one power (0)

The laws of indices can't merge these.

Where the line is: Adding or subtracting indices only works when the powers share the SAME base AND are being multiplied or divided. Lose either condition and there is nothing to count together.

8 of 8 still to sort.

Stuck on one? Write every power out as a string of factors and count them. a³ is just a·a·a.

Watch out: a² × a³ = a⁵ but (a²)³ = a⁶. They look alike, but one has five factors and the other has six. If you're ever unsure, write them out and count.

When the terms have numbers in front

Problem

Simplify (3x⁴y × 5x²y³) ÷ 3x²y

Spot the slip

Where does this answer go wrong?

Simplify (2a³b)⁴

A student's answer — which line goes wrong?

Raise each part of the bracket to the power 4.
The a-part: (a³)⁴ = a¹², because it's a power of a power, so 3 × 4 = 12.
The b-part: b is b¹, so (b¹)⁴ = b⁴.
The number part: 2 is just a coefficient, so it stays as 2.
So (2a³b)⁴ = 2a¹²b⁴

Strange indices

Where do these powers land?

Zero, negative and fractional indices look odd. Go with your instinct: drag each power to where you think its value sits, then check. (9^(1/2) means 9 to the power ½.)

Place 5⁰Place 2⁻¹Place 2⁻²9^½ — 9^(1/2)8^⅔ — 8^(2/3)Place 2³

Maths · Algebra

Why the strange indices mean what they mean

Work each one out two ways. Both answers must match, and that forces the meaning.

GoalWhy does a⁰ = 1?

a³ ÷ a³

Divide a power by itself. There are two ways to work this out.

Step 1 of 4

Divide a power by itself. There are two ways to work this out.

Watch out: a⁰ = 1 only works when a isn't 0. The argument divides by a, and you can't divide by 0.

Put it to work

Solve problems with the laws

(a) Write √x × x³ as a single power of x. (b) 3ᵏ × 3² = 3⁴ ÷ 3⁻³. Find the value of k. [5 marks]

Your answer

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The mark scheme

(a) √x written as x^(1/2) (1 mark)
(a) x^(7/2) (accept x^3.5) (1 mark)
(b) Right-hand side simplified to 3⁷, because 4 − (−3) = 7 (1 mark)
(b) Left-hand side written as 3^(k + 2), giving the equation k + 2 = 7 (1 mark)
(b) k = 5 (1 mark)

Model answer

(a) √x = x^(1/2), so √x × x³ = x^(1/2 + 3) = x^(7/2). (b) Right-hand side: 3⁴ ÷ 3⁻³ = 3^(4 − (−3)) = 3⁷. Left-hand side: 3ᵏ × 3² = 3^(k + 2). The bases match, so the indices must be equal: k + 2 = 7, so k = 5.

When every term has the same base, stop working with the powers and write an equation in the indices. From there it's ordinary algebra. And watch the signs: subtracting a negative index adds, so 4 − (−3) = 7.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Every law is just counting repeated factors, so you can rebuild any of them on the spot.

What you need to know

An index counts identical factors: a³ means a × a × a, and a²b means a × a × b. The base can be a number, a letter or a mix of the two.
Multiplication law: aᵐ × aⁿ = aᵐ⁺ⁿ. Division law: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Power of a power: (aᵐ)ⁿ = aᵐⁿ.
Adding or subtracting indices only works for powers with the same base that are being multiplied or divided. A sum like a² + a³, or a product of different bases like a² × b³, can't be merged into one power this way.
Zero, negative and fractional indices look strange, but each has one fixed meaning, chosen so that the three laws above keep working.

The big picture

The laws of indices come from counting repeated factors. Multiplying same-base powers adds the indices, dividing subtracts them, and a power of a power multiplies them. Coefficients are multiplied or divided separately, a power on a bracket applies to everything inside it, and zero, negative and fractional indices are defined (a⁰ = 1, a⁻ⁿ = 1/aⁿ, a^(1/n) = ⁿ√a) so that the laws keep working.

Key points

1Multiplying powers with the same base: add the indices, aᵐ × aⁿ = aᵐ⁺ⁿ.

2Dividing powers with the same base: subtract the indices, aᵐ ÷ aⁿ = aᵐ⁻ⁿ.

3Power of a power: multiply the indices, (aᵐ)ⁿ = aᵐⁿ.

4With coefficients, multiply or divide the numbers, and add or subtract the indices of each letter separately.

5A power outside a bracket applies to everything inside, including the coefficient.

6a⁰ = 1 (a ≠ 0), a⁻ⁿ = 1/aⁿ, a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)ᵐ.

7Surds can be written as powers, for example √x = x^(1/2), and then the same laws apply.

8To find an unknown index, write every term as a power of the same base and set the indices equal.

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Worked example

Problem

Work out the value of 27^(−2/3) without a calculator.

⚠ Watch out

Multiplying the indices when you multiply powers. a² × a³ is a⁵, not a⁶: write it out and there are only five a's. You only multiply the indices for a power of a power, such as (a²)³.

🧠

Memory hook

Times → add. Divide → subtract. Bracket power → multiply. And the odd ones: zero → 1, minus → 'one over', fraction → root (bottom is the root, top is the power). If in doubt, write the factors out and count.

✓

Check yourself

Using only written-out factors, explain why a² × a³ = a⁵ but (a²)³ = a⁶. Then say what 10⁰ equals, and why it can't be 0.

Flashcards

(14)

What does the index in a⁴ tell you?

How many identical factors are multiplied together: a⁴ = a × a × a × a.

When can you combine two powers by adding or subtracting their indices?

Only when they have the same base AND are being multiplied or divided. Not for a sum like a² + a³, and not for different bases like a² × b³.

aᵐ × aⁿ = ?

aᵐ⁺ⁿ. Same base, multiplying: add the indices.

What do you do to the indices when you divide a⁹ by a⁴?

Subtract them: a⁹⁻⁴ = a⁵. In general aᵐ ÷ aⁿ = aᵐ⁻ⁿ.

Power of a power

(aᵐ)ⁿ = aᵐⁿ: multiply the indices. It's the same count added n times.

What index does a letter on its own, like y, have?

1, because y = y¹.

Simplify 2p³ × 7p²

14p⁵: multiply the numbers, add the indices.

Simplify 20t⁹ ÷ 4t³

5t⁶: divide the numbers, subtract the indices.

What does a power outside a bracket apply to?

Everything inside, including the number: (5k²)² = 25k⁴.

a⁰ = ?

1, for any a that isn't 0.

a⁻ⁿ = ?

1/aⁿ, which means 'one over'. It is never a negative number just because the index is negative.

a^(1/n) = ?

ⁿ√a, the nth root. So a^(1/2) = √a.

a^(m/n) = ?

(ⁿ√a)ᵐ: the bottom of the fraction is the root, the top is the power.

Write √x as a power of x

x^(1/2)

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Which single power does each one make?

Where does this answer go wrong?

Where do these powers land?

Why the strange indices mean what they mean

Solve problems with the laws

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