Square
\(7^2=7\times7=49\).
A square can also be visualised as the area of a square with equal side lengths.
Chapter 3
Powers and roots
Refresh the essential square, square-root, cube and cube-root facts you need throughout Mathematics, then practise using calculators confidently for other powers and roots.
By the end of this chapter, you should be able to
- find squares and positive square roots with and without a calculator;
- recall the square numbers from \(1^2\) to \(15^2\);
- evaluate expressions containing squares and square roots;
- find cubes and cube roots with and without a calculator;
- recall key cube numbers, including \(1^3,2^3,3^3,4^3,5^3\) and \(10^3\);
- use calculator power and root functions for other powers and roots;
- round calculator answers appropriately when a question asks for a stated accuracy.
1
Subtopic 1
Squares and square roots
To square a number, multiply it by itself. The notation \(a^2\) means \(a\times a\). A square root reverses this process: \(\sqrt{a}\) is the positive number whose square is \(a\).
Key distinction: the symbol \(\sqrt{a}\) denotes the principal (non-negative) square root. But an equation such as \(x^2=25\) has two solutions, \(x=\pm5\). Work out powers before adding or subtracting.
Positive square root
\(\sqrt{49}=7\) because \(7^2=49\).
The radical symbol means “the positive square root of”.
Square numbers to know
You should be able to recall these quickly without a calculator.
| Integer | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Square | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 | 169 | 196 | 225 |
Without a calculator
For a square
- Write the multiplication explicitly if needed.
- Multiply the number by itself.
- For familiar integers, use the memorised square-number table.
For an exact square root
- Ask which positive number squares to give this value.
- Use the square-number table.
- Check by squaring your answer.
Common error: \(8^2\) is \(8\times8=64\), not \(8\times2\). Work out powers before addition or subtraction in a mixed expression.
Worked example 1 · Easy
Recall a square and a square root
Work out \(13^2\) and \(\sqrt{196}\) without a calculator.
From the square-number facts:
\[13^2=169\]and since \(14^2=196\),
\[\sqrt{196}=14.\]Worked example 2 · Easy
An expression containing squares
Work out \(9^2-5^2\).
Evaluate each power first:
\[9^2=81,\qquad5^2=25\]Then subtract:
\[81-25=\boxed{56}.\]Worked example 3 · Medium
Recognise an exact square root
Work out \(\sqrt{841}\) without using a square-root key.
We know \(30^2=900\), so the required integer is slightly below 30. Check 29:
\[29^2=841\]Therefore
\[\boxed{\sqrt{841}=29}.\]Worked example 4 · Exam style
Use a square root in context
A square patio has area \(324\text{ m}^2\). Find its side length and perimeter.
If the side length is \(s\), then \(s^2=324\).
\[s=\sqrt{324}=18\text{ m}.\]The perimeter is
\[4\times18=\boxed{72\text{ m}}.\]Additional worked example 1 · Medium
Squares and order of operations
Evaluate \(15^2-\sqrt{144}+3^2\).
Step 1: Calculate the powers and root: \(15^2=225\), \(\sqrt{144}=12\), \(3^2=9\).
Step 2: \(225-12+9=\boxed{222}\).
Additional worked example 2 · Exam-Style
A square in context
A square garden has area \(196\text{ m}^2\). A 2 m gate replaces part of its boundary. Find the length of fencing required.
Step 1: Side \(=\sqrt{196}=14\text{ m}\).
Step 2: Full perimeter \(=4\times14=56\text{ m}\).
Step 3: Fence required \(=56-2=\boxed{54\text{ m}}\).
Practice this subtopic
Attempt the early questions first, then the exam-style question. Open Show Answer only after trying your own working.
1.1
Easy [1 mark]
Find \(\sqrt{169}\).
Show Answer
Step 1: \(13^2=169\).
Step 2: \(\boxed{13}\).
1.2
Medium [2 marks]
Work out \(17^2-12^2\).
Show Answer
Step 1: Calculate the squares: 289 and 144.
Step 2: \(289-144=\boxed{145}\).
1.3
Medium [2 marks]
Find all real solutions of \(x^2=81\).
Show Answer
Step 1: Both positive and negative 9 have square 81.
Step 2: \(\boxed{x=9\text{ or }x=-9}\).
1.4
Exam-Style [3–4 marks]
A square courtyard has area \(324\text{ m}^2\). A 3 m opening is left in the perimeter. How much fencing is needed?
Show Answer
Step 1: Side \(=\sqrt{324}=18\text{ m}\).
Step 2: Perimeter \(=4\times18=72\text{ m}\).
Step 3: Fence \(=72-3=\boxed{69\text{ m}}\).
2
Subtopic 2
Cubes and cube roots
To cube a number, multiply it by itself three times. The notation \(a^3\) means \(a\times a\times a\). A cube root reverses this process: \(\sqrt[3]{a}\) is the number whose cube is \(a\).
Cubing is repeated multiplication three times: \(a^3=a\times a\times a\). Unlike square roots, cube roots of negative numbers are negative: \(\sqrt[3]{-125}=-5\). The textbook also asks you to use calculator powers other than 2 or 3.
Cube
\(4^3=4\times4\times4=64\).
A cube can be visualised as the volume of a cube whose three dimensions are equal.
Cube root
\(\sqrt[3]{64}=4\) because \(4^3=64\).
Cube-root questions may be exact or require a calculator approximation.
Cube numbers to know
| Integer | 1 | 2 | 3 | 4 | 5 | 10 |
|---|---|---|---|---|---|---|
| Cube | 1 | 8 | 27 | 64 | 125 | 1000 |
Using a calculator for powers and roots
Powers
Use the square/cube keys where available, or the general power key for results such as \(2.4^5\).
Roots
Use square-root and cube-root keys for common roots. For other roots, use your calculator’s general \(n\)th-root function. Calculator layouts differ, so learn the keys on your own model.
Keep the full calculator value during working and round the final answer only when the question asks you to.
Worked example 5 · Easy
Recall a cube and cube root
Work out \(5^3\) and \(\sqrt[3]{216}\).
\[5^3=5\times5\times5=125\]
Since \(6^3=216\),
\[\sqrt[3]{216}=\boxed{6}.\]Worked example 6 · Medium
Cube a decimal
Work out \(2.4^3\).
\[2.4^3=2.4\times2.4\times2.4=\boxed{13.824}.\]
Worked example 7 · Medium
A non-exact cube root
Work out \(\sqrt[3]{175}\), giving your answer to 2 decimal places.
\[\sqrt[3]{175}\approx5.593\ldots\]
To 2 decimal places:
\[\boxed{5.59}.\]Worked example 8 · Exam style
Find the side length of a cube
A cube has volume \(350\text{ cm}^3\). Find its side length to 2 decimal places.
If the side length is \(s\), then
\[s^3=350\]so
\[s=\sqrt[3]{350}\approx7.047\ldots=\boxed{7.05\text{ cm}}.\]Worked example 9 · Exam style
Other powers and roots
Work out \(1.6^5\) and \(\sqrt[4]{2401}\).
\[1.6^5=\boxed{10.48576}\]
and because \(7^4=2401\),
\[\sqrt[4]{2401}=\boxed{7}.\]Additional worked example 1 · Medium
Cube and cube root together
Calculate \(\sqrt[3]{512}+(-3)^3\).
Step 1: \(8^3=512\), so \(\sqrt[3]{512}=8\).
Step 2: \((-3)^3=-27\).
Step 3: \(8-27=\boxed{-19}\).
Additional worked example 2 · Exam-Style
Find a cube’s edge
A cube has a volume of \(3375\text{ cm}^3\). Find the side and the total length of its twelve edges.
Step 1: Edge \(=\sqrt[3]{3375}=15\text{ cm}\).
Step 2: Twelve equal edges total \(12\times15=\boxed{180\text{ cm}}\).
Practice this subtopic
Attempt the early questions first, then the exam-style question. Open Show Answer only after trying your own working.
2.1
Easy [1 mark]
Find \(\sqrt[3]{216}\).
Show Answer
Step 1: \(6^3=6\times6\times6=216\).
Step 2: \(\boxed6\).
2.2
Medium [2 marks]
Calculate \((-4)^3+\sqrt[3]{27}\).
Show Answer
Step 1: \((-4)^3=-64\), \(\sqrt[3]{27}=3\).
Step 2: \(-64+3=\boxed{-61}\).
2.3
Medium [2 marks]
Find \(\sqrt[3]{0.008}\).
Show Answer
Step 1: \(0.2\times0.2\times0.2=0.008\).
Step 2: \(\boxed{0.2}\).
2.4
Exam-Style [3–4 marks]
A cube-shaped tank has volume \(0.125\text{ m}^3\). Find its edge in centimetres.
Show Answer
Step 1: Edge in metres \(=\sqrt[3]{0.125}=0.5\text{ m}\).
Step 2: Convert units: \(0.5\times100=\boxed{50\text{ cm}}\).
Chapter 3 practice
Start with quick recall, then move to calculator work and exam-style applications. Try each question before opening the solution.
A. Build confidence
1Square-number recall[3 marks]
Work out \(6^2\), \(11^2\) and \(15^2\).
Show Answer
\[6^2=36,\qquad11^2=121,\qquad15^2=225.\]
2Square-root recall[3 marks]
Work out \(\sqrt{64}\), \(\sqrt{169}\) and \(\sqrt{225}\).
Show Answer
\[\boxed{8,\ 13,\ 15}.\]
3Evaluate powers first[3 marks]
Work out \(12^2-7^2\).
Show Answer
\[144-49=\boxed{95}.\]
4Cube-number recall[3 marks]
Work out \(3^3\), \(4^3\) and \(10^3\).
Show Answer
\[27,\ 64,\ 1000.\]
5Cube-root recall[3 marks]
Work out \(\sqrt[3]{8}\), \(\sqrt[3]{64}\) and \(\sqrt[3]{125}\).
Show Answer
\[\boxed{2,\ 4,\ 5}.\]
B. Develop your method
6Mixed square expression[4 marks]
Work out \(13^2-8^2+\sqrt{144}\).
Show Answer
\[169-64+12=\boxed{117}.\]
7Decimal cube[2 marks]
Work out \(1.8^3\).
Show Answer
\[1.8^3=\boxed{5.832}.\]
8Cube root to 2 decimal places[3 marks]
Work out \(\sqrt[3]{600}\), giving your answer to 2 decimal places.
Show Answer
\[\sqrt[3]{600}=8.434\ldots\]
Therefore \(\boxed{8.43}\).
9Other power[2 marks]
Use a calculator to work out \(2.3^4\).
Show Answer
\[2.3^4=\boxed{27.9841}.\]
10Other root[2 marks]
Work out \(\sqrt[4]{625}\).
Show Answer
Since \(5^4=625\),
\[\boxed{\sqrt[4]{625}=5}.\]C. Exam-style challenge
11Mixed powers and roots[4 marks]
Work out \(3^4+\sqrt{225}-\sqrt[3]{125}\).
Show Answer
\[81+15-5=\boxed{91}.\]
12Square in context[5 marks]
A square field has area \(484\text{ m}^2\). Find the side length and perimeter.
Show Answer
\[\sqrt{484}=22\text{ m}\]
so the perimeter is
\[4\times22=\boxed{88\text{ m}}.\]13Cube in context[4 marks]
A cube has volume \(90\text{ cm}^3\). Find its side length to 2 decimal places.
Show Answer
\[s=\sqrt[3]{90}=4.481\ldots\]
so \(\boxed{s=4.48\text{ cm}}\).
14Fifth root[3 marks]
A positive number \(x\) satisfies \(x^5=16807\). Find \(x\).
Show Answer
\[x=\sqrt[5]{16807}=\boxed{7}.\]
Chapter 3 high-grade checklist
Square facts
I know the square numbers from \(1^2\) to \(15^2\) and can reverse them to find exact square roots.
Cube facts
I know the key cube numbers and can reverse them to find exact cube roots.
Calculator skills
I can use my own calculator for square roots, cube roots, general powers and other roots.
Accuracy
I keep full calculator values during working and round the final answer only when requested.
Chapter 3 revision: This chapter follows the subtopic headings and core concepts in the supplied Cambridge O Level Mathematics, Second Edition. Figure crops come from the provided textbook. Additional questions are original exam-style practice, not reproduced Cambridge examination questions.