Powers and Roots
Master powers and roots for GCSE Maths with structured, exam-style practice. This Foundation resource covers powers and roots through a range of exam-style questions and includes a complete mark scheme showing the steps examiners reward. Suitable for AQA, Edexcel and OCR. Free to download as a PDF. Squaring and square-rooting are inverse operations, as are cubing and cube-rooting.
Free downloads
These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.
Challenge / Extension
Stretch yourself beyond the basics.
Topic overview
Powers and roots extend squaring and square rooting to any index. A power tells you how many times to multiply a number by itself, so \(2^5\) means five twos multiplied together, giving \(32\).
Cube numbers are worth knowing alongside squares: \(1, 8, 27, 64\) and \(125\) are the cubes of \(1\) to \(5\). The cube root reverses cubing, so the cube root of \(64\) is \(4\).
Unlike square roots, cube roots work for negative numbers. Because a negative multiplied three times stays negative, the cube root of \(-8\) is \(-2\). A negative number has no real square root, which is a distinction worth remembering.
Revision notes
Powers
\(a^n\) means \(a\) multiplied by itself \(n\) times. So \(3^4 = 3 \times 3 \times 3 \times 3 = 81\).
Be careful with negatives: \((-2)^4 = 16\) because the negatives pair up, but \((-2)^3 = -8\) because one negative is left over. An even power gives a positive result; an odd power keeps the sign.
Cubes and cube roots
A cube number is \(n^3\). The first five are \(1, 8, 27, 64, 125\).
The cube root reverses this, written \(\sqrt[3]{\ }\), so \(\sqrt[3]{125} = 5\). Cube roots of negative numbers exist: \(\sqrt[3]{-27} = -3\).
Higher roots and estimation
The fourth root undoes the fourth power, and so on. \(\sqrt[4]{16} = 2\) because \(2^4 = 16\).
When a root is not exact, locate it between the two nearest whole-number powers, just as with square roots.
Key points
- \(a^n\) means \(a\) multiplied by itself \(n\) times.
- Cube numbers: 1, 8, 27, 64, 125.
- An even power of a negative number is positive.
- An odd power of a negative number is negative.
- Cube roots of negative numbers exist.
- Negative numbers have no real square root.
Worked examples
Example 1
Work out \(2^6\).
Working
Example 2
Work out \(\sqrt[3]{216}\).
Working
Example 3
Work out \((-3)^4\) and \((-3)^3\).
Working
Common mistakes
Multiplying the base by the index.
2⁵ is 32, not 10. The index counts how many times the base multiplies itself.
Getting the sign wrong with negative bases.
(−2)⁴ is +16 but (−2)³ is −8. Count whether the power is odd or even.
Assuming negative numbers have square roots.
There is no real square root of −9, but the cube root of −27 is −3.
Confusing \(-3^2\) with \((-3)^2\).
Without brackets only the 3 is squared, giving −9. With brackets the answer is 9.
Exam tips
- Learn the cubes of 1 to 5 alongside the squares.
- Check whether the index is odd or even when the base is negative.
- Use brackets around a negative base so the sign is unambiguous.
- Verify a root by raising your answer back to the power.
Key terms
- Power
- How many times a base is multiplied by itself.
- Cube number
- The result of multiplying a whole number by itself three times.
- Cube root
- The value that cubes to give a number.
- Real number
- A number that exists on the ordinary number line.
Related topics
Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.