Squaring and cubing build numbers up; square roots and cube roots take them back down. Know the squares to 15 and the key cubes by heart, then use a calculator with confidence for the rest.
A root undoes a power: \(\sqrt{169} = 13\) because \(13^{2} = 169\), and \(\sqrt[3]{64} = 4\) because \(4^{3} = 64\). A square root of a positive number is taken as positive; a cube root keeps the sign, so \(\sqrt[3]{-125} = -5\).
Fig. 1: a square of side 4 is made of 4² = 16 unit squares; a cube of edge 2 is made of 2³ = 8 unit cubes. The root recovers the side from the total
Section 01
Squares and square roots.
In the lab, slide \(n\) and watch the square grow as \(n^{2}\) unit squares; then switch to roots, type a number and see which two squares it lies between. Say each square aloud until you know 1 to 15 without the lab.
\(n\)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
\(n^{2}\)
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
Worked example Recall
Write down the value of (a) \(\sqrt{169}\) (b) \(\sqrt{0.49}\).
(a) \(13^{2} = 169\), so \(\sqrt{169} = 13\).
(b) \(0.7^{2} = 0.49\) (since \(7^{2} = 49\) and \(0.7 \times 0.7\) has two decimal places).
Answer: (a) 13 (b) 0.7
Worked example Trapping a root
Without a calculator, find the two consecutive integers that \(\sqrt{50}\) lies between.
Find the squares either side of 50: \(7^{2} = 49\) and \(8^{2} = 64\).
Answer: \(7 < \sqrt{50} < 8\) (and it is only just above 7)
Examiner trap
\(-3^{2}\) means \(-(3^{2}) = -9\), but \((-3)^{2} = 9\). On a calculator, put brackets round a negative number before squaring it.
Section 02
Cubes and cube roots.
You must recall the cubes of 1, 2, 3, 4, 5 and 10 and their cube roots. Unlike square roots, cube roots of negative numbers exist: a negative times a negative times a negative is negative.
\(n\)
1
2
3
4
5
10
\(n^{3}\)
1
8
27
64
125
1000
Worked example Without a calculator
Work out \(5^{2} \times \sqrt[3]{8}\).
\(5^{2} = 25\).
\(\sqrt[3]{8} = 2\) because \(2^{3} = 8\).
\(25 \times 2 = 50\).
Answer: 50
Worked example Negative and decimal cube roots
Write down (a) \(\sqrt[3]{-125}\) (b) \(\sqrt[3]{0.008}\).
Cubing is not multiplying by 3: \(4^{3} = 4 \times 4 \times 4 = 64\), not 12. In the same way \(\sqrt[3]{27} = 3\), not 9.
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
Write down \(\sqrt{144}\).
Quick check+10 XP
Work out \(4^{3}\).
Quick check+10 XP
Write down \(\sqrt[3]{-27}\).
Quick check+10 XP
Which of these lies between 6 and 7?
Section 03
Other powers and roots, and the calculator.
\(2^{5} = 2 \times 2 \times 2 \times 2 \times 2 = 32\), so the fifth root of 32 is 2: \(\sqrt[5]{32} = 2\). For harder values use the calculator's \(x^{\square}\) and \(\sqrt[\square]{\ }\) keys, and put brackets round anything under a root sign that has more than one term.
Worked example Roots of a calculation
Work out \(\sqrt{12.5^{2} - 3.5^{2}}\).
Work out under the root first: \(12.5^{2} = 156.25\) and \(3.5^{2} = 12.25\).
\(156.25 - 12.25 = 144\).
\(\sqrt{144} = 12\).
Answer: 12
Worked example Higher powers
Work out (a) \(3^{4}\) (b) \(\sqrt[4]{81}\) (c) \(1.3^{3}\).
(a) \(3 \times 3 \times 3 \times 3 = 81\).
(b) The fourth root undoes the fourth power, and \(3^{4} = 81\).
\(\sqrt{12.5^{2} - 3.5^{2}}\) is NOT \(12.5 - 3.5 = 9\). A root does not split over a plus or minus sign; work out the whole expression underneath first.
Stage 2 · Exam
Exam-style questions
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Finish the four checks above to unlock the exam questions
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Without a calculator, work out \(5^{2} \times \sqrt[3]{8}\).
Exam style+20 XP
Work out \(\sqrt{13^{2} - 5^{2}}\).
Exam style+20 XP
Use your calculator to work out \(\dfrac{\sqrt{20.25} + 1.5^{3}}{2^{4}}\). Give your answer correct to 3 significant figures.
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A cube has volume 343 cm3. Find its total surface area.
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