Powers, roots and estimating

Powers and roots undo each other. Knowing the small powers by heart lets you estimate roots without a calculator.

Practise this lesson free (7 questions)

Powers to know

25 = 32, 210 = 1024
34 = 81, 43 = 64, 53 = 125
A negative number to an odd power stays negative: (−2)3 = −8. To an even power it becomes positive: (−2)4 = 16.
n12345
2n2481632
3n392781243
4n416642561024
5n5251256253125

Roots

A root undoes a power. √49 = 7 because 72 = 49.
∛{125} = 5 because 53 = 125. ∛{−8} = −2, because (−2)3 = −8.
Higher roots work the same way: ⁴√81 = 3 because 34 = 81.
The reciprocal of a number is 1 divided by it: the reciprocal of 4 is 14 = 0.25.

Estimating a root

√70 is between √64 = 8 and √81 = 9, and closer to 8.
So √70 ≈ 8.4 (a calculator gives 8.367...).
In calculations follow BIDMAS: powers and roots come before × and ÷.
2 + 3 × 42 = 2 + 3 × 16 = 50
Worked example

Estimate √38 to 1 decimal place without a calculator.

  1. 36 < 38 < 49, so √38 is between 6 and 7.
  2. 38 is just above 36, so the root is just above 6.
  3. 6.12 = 37.21 and 6.22 = 38.44, so √38 ≈ 6.2

Answer: About 6.2 (a calculator gives 6.164...)

Key idea

Learn the powers of 2, 3, 4 and 5. A root undoes a power. Estimate a root from the square numbers either side of it.

Key words
Estimate
An approximate answer, usually found by rounding each number to 1 significant figure.
Reciprocal
1 divided by a number. The reciprocal of 4 is 1/4; a number times its reciprocal is 1.

All GCSE and International GCSE Mathematics key terms and definitions →

Common mistakes
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Extra practice

2 more questions

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  1. Work out 55

    Show the answer

    Answer: 3125

    5 × 5 × 5 × 5 × 5 = 3125.

  2. Work out ∛{1000}

    Show the answer

    Answer: 10

    10 × 10 × 10 = 1000.

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Sources and references

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

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