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The square root of 144 is 12 because 12 × 12 = 144, and the cube root of 125 is 5. Free GCSE AQA Maths notes with worked examples and 13 questions answered.
Powers and Roots is part of Structure and calculation in GCSE AQA Higher Maths.
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GCSE AQA Higher
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The square root of 144 is 12 because 12 × 12 = 144, and the cube root of 125 is 5. Free GCSE AQA Maths notes with worked examples and 13 questions answered.
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Use this lesson to stay inside Structure and calculation while connecting the detail back to the wider Number topic.
Read the core explanations from Powers and Roots before testing yourself from memory.
Every time you calculate an area, work with very large or very small numbers, or use a formula in science, powers and roots are doing the heavy lifting behind the scenes. Mastering them now makes everything from algebra to standard form far more manageable.
The core idea is compact: instead of writing $5 \times 5 \times 5 \times 5$, you write $5^4$. That superscript number — called the index or power — tells you how many times the base is multiplied by itself. Roots reverse that process: they ask "what number was multiplied by itself to give this result?"
The index tells you the number of times the base appears in the multiplication — not the number of multiplications performed. For $2^3$, the base 2 appears three times: $2 \times 2 \times 2 = 8$. There are only two multiplication signs, but three factors.
Two powers have special names you must know: $a^2$ is called "$a$ squared" and $a^3$ is called "$a$ cubed". For higher powers such as $a^4$ and $a^5$, you say "$a$ to the power of 4" and "$a$ to the power of 5".
The higher-tier specification asks you to recognise powers of 2, 3, 4, and 5. The table below gives you the values you are expected to know without a calculator. Commit these to memory — they appear constantly in questions without a calculator.
| Power | Powers of 2 | Powers of 3 | Powers of 4 | Powers of 5 |
|---|---|---|---|---|
| $n = 1$ | $2^1 = 2$ | $3^1 = 3$ | $4^1 = 4$ | $5^1 = 5$ |
| $n = 2$ | $2^2 = 4$ | $3^2 = 9$ | $4^2 = 16$ | $5^2 = 25$ |
| $n = 3$ | $2^3 = 8$ | $3^3 = 27$ | $4^3 = 64$ | $5^3 = 125$ |
| $n = 4$ | $2^4 = 16$ | $3^4 = 81$ | $4^4 = 256$ | $5^4 = 625$ |
| $n = 5$ | $2^5 = 32$ | $3^5 = 243$ | $4^5 = 1024$ | $5^5 = 3125$ |
When working out a power by hand, expand it fully as a chain of multiplications. Never try to do it all in one mental leap — that is where errors creep in. Work left to right, one multiplication at a time.
Worked example: Calculating a Power by Repeated Multiplication
Work out the value of $4^3$.
Answer: $4^3 = 64$
Worked example: Calculating a Higher Power
Work out the value of $3^5$.
Answer: $3^5 = 243$
A root is the inverse operation of a power. If $7^2 = 49$, then $\sqrt{49} = 7$. You are asking: "what number, raised to this power, gives me the result inside the root symbol?"
Higher roots follow the same pattern. The fourth root $\sqrt[4]{n}$ asks for the number that, raised to the power of 4, gives $n$. For example, $\sqrt[4]{81} = 3$ because $3^4 = 3 \times 3 \times 3 \times 3 = 81$.
Worked example: Finding a Square Root Without a Calculator
Find $\sqrt{144}$.
Answer: $\sqrt{144} = 12$
Worked example: Finding a Cube Root Without a Calculator
Find $\sqrt[3]{125}$.
Answer: $\sqrt[3]{125} = 5$
The relationship between powers and roots is a two-way street. Knowing your powers table makes finding roots much faster — you are reading the table in reverse. If you know $4^3 = 64$, you immediately know $\sqrt[3]{64} = 4$.
Powers and roots questions on GCSE Higher exam papers are often short — one or two marks — but they are dropped surprisingly often. The mistakes below are the ones that appear most frequently in mark scheme reports. Knowing them in advance puts you ahead.
Powers and roots become much safer once you connect them to inverse operations and estimation. Squaring and square rooting undo each other, cubing and cube rooting undo each other, and negative powers can be rewritten as fractions to make the structure clearer. In exam questions, estimate the size of the answer before calculating exactly: for example, knowing that 7 squared is 49 and 8 squared is 64 helps you judge whether a square root or index result is sensible. This habit catches arithmetic slips quickly and strengthens your understanding of how powers change scale.
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Move to the related lessons or back to the topic page when you need the wider sequence before exam-style practice.
The square root of 144 is 12, because 12 × 12 = 144. This can be checked by expanding: 12 × 10 + 12 × 2 = 120 + 24 = 144.
The first five powers of 2 are 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16 and 2⁵ = 32. Powers of 2, 3, 4 and 5 are expected to be known without a calculator.
The cube root of 125 is 5, because 5³ = 5 × 5 × 5 = 125. Working it out means finding x such that x³ = 125: 5 × 5 = 25, then 25 × 5 = 125.
An index, or power, shows how many times the base appears as a factor in a repeated multiplication. For example, 5⁴ = 5 × 5 × 5 × 5, and 2³ = 2 × 2 × 2 = 8, which has three factors but only two multiplication signs.
3 to the power of 5 is 243. Write it as 3 × 3 × 3 × 3 × 3 and multiply left to right: 3 × 3 = 9, 9 × 3 = 27, 27 × 3 = 81 and 81 × 3 = 243.
The first powers of 3 are 3¹ = 3, 3² = 9, 3³ = 27, 3⁴ = 81 and 3⁵ = 243. Each is found by multiplying the one before by 3, and knowing them helps find roots such as the fourth root of 81.
4 cubed, 4³, is 64. It is worked out as 4 × 4 × 4: first 4 × 4 = 16, then 16 × 4 = 64, so the cube root of 64 is 4.
The first powers of 5 are 5¹ = 5, 5² = 25, 5³ = 125, 5⁴ = 625 and 5⁵ = 3125. Recognising these without a calculator makes roots faster, for example the cube root of 125 is 5.
The first powers of 4 are 4¹ = 4, 4² = 16, 4³ = 64, 4⁴ = 256 and 4⁵ = 1024. Knowing 4³ = 64 means the cube root of 64 is immediately 4.
The fourth root of 81 is 3, because 3⁴ = 3 × 3 × 3 × 3 = 81. A fourth root asks for the number that, raised to the power of 4, gives the number inside the root.
To find a cube root without a calculator, rewrite it as a power equation and test a sensible value. For the cube root of 125, find x such that x³ = 125; since 5 × 5 = 25 and 25 × 5 = 125, the cube root of 125 is 5.
Squared means raised to the power of 2 and cubed means raised to the power of 3, so a² is 'a squared' and a³ is 'a cubed'. Higher powers such as a⁴ and a⁵ are read as 'a to the power of 4' and 'a to the power of 5'.
A root is the inverse operation of a power: it asks what number, raised to a given power, gives the result inside the root symbol. If 7² = 49 then √49 = 7, and the fourth root of 81 is 3 because 3⁴ = 81.