Squares, Square Roots and Surds
Squares and square roots are the starting point for Pythagoras' theorem. This NSW Year 9 Core maths subtopic revises squaring and taking square roots, then meets surds: square roots such as \(\sqrt{57}\) that never come out exactly. Surds reach a little beyond the Core outcomes and come from NESA outcome MA5‑IND‑P‑02.
You will square whole numbers and decimals such as \(2.5^2\), find square roots of perfect squares like \(\sqrt{1369}\), evaluate expressions such as \(\sqrt{5^2 + 12^2}\) by working under the root sign first, round to one decimal place, decide whether an answer is rational or a surd, and place \(\sqrt{90}\) between two whole numbers without a calculator. These are the same calculations you use to find a missing side in Pythagoras' theorem. It is part of Year 9 Core Pythagoras Theorem.
Theory
Squaring a number multiplies it by itself, and a square root undoes it. This Year 9 maths page shows how to square whole numbers and decimals, find square roots, work out expressions like \(\sqrt{a^2 + b^2}\), decide whether a root is rational or a surd, and estimate a root without a calculator. These are the number skills behind every Pythagoras' theorem question.
To square a number, multiply it by itself: \(n^2 = n \times n\). So \(11^2 = 11 \times 11 = 121\). Squaring is not doubling.
The square root undoes squaring. \(\sqrt{n}\) asks: which positive number, times itself, gives \(n\)? Because \(11 \times 11 = 121\), \(\sqrt{121} = 11\).
A perfect square is a whole number that is another whole number squared, such as \(1\), \(4\), \(9\), \(16\) and \(25\). The square root of a perfect square is a whole number.
A number is rational if it can be written as a fraction. Whole numbers, decimals that stop and decimals that repeat are all rational. A surd is a root whose decimal never ends and never repeats, such as \(\sqrt{2}\) or \(\sqrt{110}\). It cannot be written as a fraction.
A surd is an exact value. A rounded decimal, such as \(\sqrt{2} \approx 1.41\), is only an approximation of it. Surds come up again later if you go on to Advanced maths. In this subtopic you only need to recognise one.
Squaring and its opposite:
Squaring and taking the square root undo each other. Square a positive number, then take the root, and you get back where you started.
The root sign works like a bracket. Work out what is underneath first:
How to evaluate a square root expression
- Work under the root sign first. Square each number, then add or subtract.
- Check whether the result is a perfect square: try a whole number times itself.
- Take the square root. A perfect square gives a whole number. Otherwise the exact answer is a surd: keep the root sign.
- Round only if the question asks for decimal places. Use a calculator, then round once at the end.
- Check by squaring your answer. It should give the number under the sign, or be close to it.
(i) \(13^2\)
(ii) \(\sqrt{441}\)
(i) Squaring means multiplying the number by itself, not doubling it.
| \(13^2\) | \(=\) | \(13 \times 13\) |
| \(\) | \(=\) | \(\mathbf{169}\) |
(ii) Look for the number that gives \(441\) when it is multiplied by itself. \(20^2 = 400\), so the root is a bit more than \(20\). \(441\) ends in \(1\), so try \(21\):
| \(21 \times 21\) | \(=\) | \(441\) |
| \(\therefore\ \sqrt{441}\) | \(=\) | \(\mathbf{21}\) |
Check a root by squaring the answer. It should give the number under the sign.
The root sign works like a bracket. Work out everything underneath it first.
| \(\sqrt{9^2 + 40^2}\) | \(=\) | \(\sqrt{81 + 1600}\) |
| \(\) | \(=\) | \(\sqrt{1681}\) |
| \(\) | \(=\) | \(\mathbf{41}\) |
Check: \(41 \times 41 = 1681\). The answer is not \(9 + 40 = 49\). A root does not split over a plus sign.
(i) exactly
(ii) correct to one decimal place.
Work under the root first:
| \(11^2 - 5^2\) | \(=\) | \(121 - 25\) |
| \(\) | \(=\) | \(96\) |
(i) \(96\) is not a perfect square: it sits between \(81\) and \(100\). Its root never ends or repeats, so the exact answer keeps the root sign. The answer is \(\boldsymbol{\sqrt{96}}\), a surd.
(ii) Use a calculator, then round:
| \(\sqrt{96}\) | \(=\) | \(9.7979\ldots\) |
| \(\) | \(\approx\) | \(\mathbf{9.8}\) |
\(9.8\) is an approximation. Only \(\sqrt{96}\) is exact.
(i) Without a calculator, between which two whole numbers does \(\sqrt{200}\) lie?
(ii) Evaluate \(1.4^2\). Is \(\sqrt{1.96}\) rational or a surd?
(i) Trap \(200\) between the perfect squares either side, then take the root of all three:
| \(196\) | \(\lt\) | \(200 \lt 225\) |
| \(14^2\) | \(\lt\) | \(200 \lt 15^2\) |
| \(14\) | \(\lt\) | \(\sqrt{200} \lt 15\) |
\(\sqrt{200}\) lies between 14 and 15.
(ii) Multiply the decimal by itself. One decimal place times one decimal place gives two:
| \(1.4^2\) | \(=\) | \(1.4 \times 1.4\) |
| \(\) | \(=\) | \(\mathbf{1.96}\) |
So \(\sqrt{1.96} = 1.4 = \dfrac{14}{10}\). It can be written as a fraction, so it is rational, not a surd.
Common pitfalls
Frequently asked questions
What is a square root?
The square root of a number is the positive number that gives it when multiplied by itself. \(\sqrt{121} = 11\), because \(11 \times 11 = 121\).
What is a perfect square?
A perfect square is a whole number that is another whole number squared, such as \(1, 4, 9, 16\) and \(25\). Its square root is a whole number.
What is a surd?
A surd is a root that cannot be written as a fraction. Its decimal never ends and never repeats. \(\sqrt{2}\) and \(\sqrt{110}\) are surds. \(\sqrt{81}\) is not, because it equals \(9\).
Is the square root of a number squared always the number?
Yes, for a positive number. Squaring and taking the square root undo each other, so \(\sqrt{5^2} = \sqrt{25} = 5\).
How do you estimate a square root without a calculator?
Find the perfect squares just below and just above the number. For \(20\), these are \(16\) and \(25\). So \(\sqrt{20}\) lies between \(4\) and \(5\).