Square Root and Cube Root
Every question with a fully worked solution.
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Theory
Taking a square root undoes squaring, and taking a cube root undoes cubing: \(16^2 = 256\), so \(\sqrt{256} = 16\); \(7^3 = 343\), so \(\sqrt[3]{343} = 7\). This Year 7 maths page shows how to find roots, estimate roots that are not whole numbers and use roots with other operations.
A square number is a whole number multiplied by itself: \(256 = 16 \times 16\). The square root goes backwards: \(16^2 = 256\), so \(\sqrt{256} = 16\). The sign \(\sqrt{\ }\) gives the positive root.
A cube number is a whole number used three times in a multiplication: \(343 = 7 \times 7 \times 7\). The cube root goes backwards: \(7^3 = 343\), so \(\sqrt[3]{343} = 7\).
\(\sqrt{\ }\) and \(\sqrt[3]{\ }\) are different: \(\sqrt{729} = 27\) because \(27 \times 27 = 729\), but \(\sqrt[3]{729} = 9\) because \(9 \times 9 \times 9 = 729\).
Larger roots. Estimate using multiples of \(10\), use the last digit (see Key facts), then check by multiplying.
Not a square number? Find the square numbers just below and just above: \(36 < 40 < 49\), so \(6 < \sqrt{40} < 7\). A calculator gives \(\sqrt{40} \approx 6.32\), an approximate value. Cube roots work the same way: \(27 < 50 < 64\), so \(3 < \sqrt[3]{50} < 4\).
Roots of a product. \(\sqrt{16 \times 9} = \sqrt{16} \times \sqrt{9} = 4 \times 3 = 12\). Cube roots split the same way: \(\sqrt[3]{3 \times 3 \times 3 \times 4 \times 4 \times 4} = 3 \times 4 = 12\). Work out roots first, like powers.
Last digits:
| Square ends in | Root ends in |
|---|---|
| 0 | 0 |
| 1 | 1 or 9 |
| 4 | 2 or 8 |
| 5 | 5 |
| 6 | 4 or 6 |
| 9 | 3 or 7 |
A square number never ends in \(2\), \(3\), \(7\) or \(8\). A cube number ending in \(0, 1, 4, 5, 6\) or \(9\) has a cube root ending in the same digit; a cube ending in \(2\) has a root ending in \(8\) (and \(8 \to 2\)), and a cube ending in \(3\) has a root ending in \(7\) (and \(7 \to 3\)).
How to find a larger root
- Estimate between multiples of \(10\) (\(20^2 = 400\), \(30^2 = 900\); \(10^3 = 1000\), \(20^3 = 8000\)).
- Use the last digit to choose the possible last digit of the root.
- Check by multiplying.
How to estimate a root between whole numbers
- Find the square (or cube) numbers just below and just above.
- Take their roots: the answer is between them.
- Compare the distances to see which it is closer to.
(i) \(\sqrt{144}\)
(ii) \(\sqrt{5 \times 5 \times 3 \times 3}\)
| \(\sqrt{12 \times 12}\) | \(=\) | \(12\) |
| \(\sqrt{5 \times 5} \times \sqrt{3 \times 3}\) | \(=\) | \(5 \times 3 = 15\) |
(i) \(12\)
(ii) \(15\)
(i) \(\sqrt[3]{27}\)
(ii) \(\sqrt[3]{2 \times 2 \times 2 \times 5 \times 5 \times 5}\)
| \(\sqrt[3]{3 \times 3 \times 3}\) | \(=\) | \(3\) |
| \(\sqrt[3]{2 \times 2 \times 2}\) | \(=\) | \(2\) |
| \(\sqrt[3]{5 \times 5 \times 5}\) | \(=\) | \(5\) |
| \(2 \times 5\) | \(=\) | \(10\) |
(i) \(3\)
(ii) \(10\)
(i) \(\sqrt{121}\)
(ii) \(\sqrt[3]{216}\)
(iii) Is \(50\) a square number?
(i) \(11\)
(ii) \(6\)
(iii) No: \(7^2 = 49\) and \(8^2 = 64\), and \(50\) is between them.
(i) \(\sqrt{576}\)
(ii) \(\sqrt[3]{1728}\)
| \(24 \times 24\) | \(=\) | \(576\) |
| \(12 \times 12 \times 12\) | \(=\) | \(144 \times 12 = 1728\) |
(i) Between \(20\) and \(30\); ends in \(4\) or \(6\); \(25^2 = 625\) is too big, so \(24\).
(ii) Between \(10\) and \(20\); \(1728\) ends in \(8\), so the root ends in \(2\): \(12\).
(i) Between which two whole numbers is \(\sqrt{75}\)?
(ii) Which one is it closer to?
| \(75 - 64\) | \(=\) | \(11\) |
| \(81 - 75\) | \(=\) | \(6\) |
(i) \(64 < 75 < 81\), so \(8 < \sqrt{75} < 9\).
(ii) Closer to \(9\) (\(\sqrt{75} \approx 8.66\)).
| \(3 \times 7 + 2 \times 8\) | \(=\) | \(21 + 16\) |
| \(\) | \(=\) | \(37\) |
\(37\). Check: \(21 \times 21 = 441 = 9 \times 49\).
Common pitfalls
Frequently asked questions
What is a square root?
The number that, multiplied by itself, gives the number: \(\sqrt{49} = 7\).
What is a cube root?
The number used three times in a multiplication to give the number: \(\sqrt[3]{343} = 7\).
How do you estimate a square root?
Find the square numbers either side of it.
Can a square number end in 7?
No. Square numbers never end in \(2\), \(3\), \(7\) or \(8\).
Is the square root of a product the product of the roots?
Yes: \(\sqrt{16 \times 9} = 4 \times 3 = 12\).