KS3 Maths · Topic guide

Roots, Powers and Exact Values

A square root of a number is a value that, when multiplied by itself, gives that number, written x, while a cube root, written cbrt(x), is a value that when multiplied by itself three times gives the number. Square numbers, such as 1, 4, 9, 16 and 25, and cube numbers, such as 1, 8, 27 and 64, have exact whole-number roots, but most numbers do not, so their roots must be estimated as a decimal or left as an exact value called a surd, for example 12 simplifies to 23.A square root of a number is a value that, when multiplied by itself, gives that number, written sqrt(x), while a cube root, written cbrt(x), is a value that when multiplied by itself three times gives the number. Square numbers, such as 1, 4, 9, 16 and 25, and cube numbers, such as 1, 8, 27 and 64, have exact whole-number roots, but most numbers do not, so their roots must be estimated as a decimal or left as an exact value called a surd, for example sqrt(12) simplifies to 2sqrt(3).

Year 7-9 (KS3)NumberNational Curriculum

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Before you start

Make sure you're comfortable with these topics first:

Number: Integers and Powers Indices and the Index Laws

Method

Learn the square numbers up to 15^2 = 225 and the cube numbers up to 5^3 = 125, so exact roots can be recognised instantly.
To find the square root of a perfect square, ask what number multiplied by itself gives that value.
To estimate the square root of a number that is not a perfect square, find the two consecutive perfect squares it lies between, then state the two whole numbers the root lies between.
To simplify a surd such as sqrt(40), find the largest square factor of the number, split the root into the product of two roots, and take the square root of the square factor.
Keep an answer as an exact value (a surd or a fraction) rather than a rounded decimal whenever a question asks for the exact value, since rounding loses accuracy.
Remember that finding a root is the inverse of raising to a power, so a square root undoes squaring and a cube root undoes cubing.

Worked example

(a) Estimate sqrt(40), stating the two consecutive whole numbers it lies between. (b) Simplify sqrt(40) to give an exact value.

Find the perfect squares either side of 40: 6^2 = 36 and 7^2 = 49.
Since 36 < 40 < 49, sqrt(40) lies between 6 and 7.
To simplify, find the largest square factor of 40: 40 = 4 x 10, and 4 is a perfect square.
Split the root: sqrt(40) = sqrt(4) x sqrt(10).
Take the square root of the square factor: sqrt(4) = 2, so sqrt(40) = 2sqrt(10), which is the exact value.

Practice questions

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Q1Show answer

Answer: 9

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Q2Work out cbrt(125).Show answer

Answer: 5

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Q3Show answer

Answer: 1

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Q4Show answer

Answer: 8 and 9 (since 8^2 = 64 and 9^2 = 81, and 64 < 70 < 81)

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Q5Show answer

Answer: 3sqrt(2) (18 = 9 x 2, and sqrt(9) = 3)

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Q6Show answer

Answer: 5sqrt(3) (75 = 25 x 3, and sqrt(25) = 5)

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Q7Show answer

Answer: 6 (cbrt(8) = 2, sqrt(16) = 4, and 2 + 4 = 6)

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Exam-style questions

Written in the style of a KS3 Maths exam paper, with a full mark scheme.

Q1[2 marks]

Between which two consecutive whole numbers does sqrt(130) lie? Explain how you know.

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identifying the two consecutive perfect squares either side of 130, 121 (= 11^2) and 144 (= 12^2)1 mark
the final answer, between 11 and 12, with the explanation that 121 < 130 < 1441 mark

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Q2[3 marks]

Simplify sqrt(200), giving your answer as an exact value in the form asqrt(b), where b has no square factors.

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identifying the largest square factor of 200, which is 1001 mark
splitting the root correctly, sqrt(100) x sqrt(2)1 mark
the fully simplified exact value, 10sqrt(2) (2 has no square factors, so this is fully simplified)1 mark

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Q3[4 marks]

A square garden has an area of 45 m^2. (a) Explain why the side length of the garden is not a whole number of metres. (b) Give the exact side length of the garden as a simplified surd.

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recognising that 45 is not a perfect square (it lies between the perfect squares 36 and 49)1 mark
a correct explanation, e.g. since 45 is not a perfect square, sqrt(45) is not a whole number1 mark
identifying the largest square factor of 45, which is 9, and splitting the root, sqrt(9) x sqrt(5)1 mark
the final exact side length, 3sqrt(5) metres1 mark

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Free printable worksheet

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Next topics

Pythagoras' Theorem Introduction to Trigonometry Introduction to Factorising Quadratics

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