Simplifying Surds
Every question with a fully worked solution.
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Theory
To simplify a surd, split the number under the root into its largest square factor times what is left, then take the square root of the square: \(\sqrt{108} = 6\sqrt{3}\). This Year 10A maths page covers entire surds, ordering, squaring a surd, cube roots, fractions and decimals under a root, pronumerals and exact answers to measurement problems.
A surd is a root that cannot be written as a fraction, such as \(\sqrt{2}\), \(\sqrt{50}\) or \(\sqrt[3]{4}\). Its decimal goes on forever without repeating, so the surd itself is the exact value. \(\sqrt{49} = 7\) is not a surd.
Surds are simplified with the rule \(\sqrt{ab} = \sqrt{a}\times\sqrt{b}\) for \(a, b \ge 0\). Choose \(a\) to be the largest square factor: \(\sqrt{108} = \sqrt{36}\times\sqrt{3} = 6\sqrt{3}\). A surd is in simplest form when the number under the root has no square factor except \(1\).
Going the other way, \(6\sqrt{3}\) can be written as the entire surd \(\sqrt{108}\), by squaring the \(6\) and moving it inside. Entire surds are easy to compare: the larger the number under the root, the larger the surd.
Cube roots work the same way with cube factors (\(8, 27, 64, 125, \ldots\)). Surds are used constantly in senior mathematics in Years 11 and 12.
| Task | Rule, with an example |
|---|---|
| Simplify | \(\sqrt{k^2m} = k\sqrt{m}\) e.g. \(\sqrt{200} = 10\sqrt{2}\) |
| Entire surd | \(k\sqrt{m} = \sqrt{k^2m}\) e.g. \(2\sqrt{7} = \sqrt{28}\) |
| Square | \((k\sqrt{m})^2 = k^2m\) e.g. \((2\sqrt{7})^2 = 28\) |
| Cube root | \(\sqrt[3]{k^3m} = k\sqrt[3]{m}\) e.g. \(\sqrt[3]{24} = 2\sqrt[3]{3}\) |
| Pronumerals | \(\sqrt{x^2} = x\) for \(x\ge 0\) e.g. \(\sqrt{x^5} = x^2\sqrt{x}\) |
| Prime powers | pair up even powers e.g. \(\sqrt{2^3\times 7^2} = 2\times 7\sqrt{2} = 14\sqrt{2}\) |
How to simplify a surd
- Find the largest square number that divides the number under the root. Work down the list \(4, 9, 16, 25, 36, 49, 64, 81, 100, \ldots\).
- Split the root: \(\sqrt{252} = \sqrt{36}\times\sqrt{7}\).
- Root the square and multiply by any number already outside.
- Check that what is left under the root has no square factor. If you used a smaller square, simplify again.
Other question types
- Surd over a number: simplify the surd first, then cancel the whole numbers, never a number under the root with one outside.
- Mixed numerals and decimals: write them as fractions, \(1\dfrac{7}{9} = \dfrac{16}{9}\) or \(0.12 = \dfrac{12}{100}\), then root the top and the bottom.
- Pronumerals: split each power into an even power times what is left, \(a^5 = a^4\times a\), and take half of each even index.
- Prime power form: with \(\sqrt{2^m\times 5^n}\), pairs of equal primes come out. To work backwards, write the answer as an entire surd and factorise the number into primes.
- Exact lengths: Pythagoras' theorem, circle areas and volumes often give a surd. Simplify it, and use a calculator only if a decimal answer is asked for.
(i) \(\sqrt{252}\)
(ii) \(\dfrac{5\sqrt{98}}{14}\)
(iii) \(\sqrt{45a^4b^3}\), where \(b\gt 0\)
(i) The largest square factor of \(252\) is \(36\).
| \(\sqrt{252}\) | \(=\) | \(\sqrt{36}\times\sqrt{7}\) |
| \(\) | \(=\) | \(\mathbf{6\sqrt{7}}\) |
(ii) Simplify the surd, then cancel the whole numbers.
| \(\dfrac{5\sqrt{98}}{14}\) | \(=\) | \(\dfrac{5\times 7\sqrt{2}}{14}\) |
| \(\) | \(=\) | \(\mathbf{\dfrac{5\sqrt{2}}{2}}\) |
(iii) Take out every square: \(45 = 9\times 5\), \(a^4 = (a^2)^2\), \(b^3 = b^2\times b\).
| \(\sqrt{45a^4b^3}\) | \(=\) | \(\sqrt{9}\times\sqrt{a^4}\times\sqrt{b^2}\times\sqrt{5b}\) |
| \(\) | \(=\) | \(\boldsymbol{3a^2b\sqrt{5b}}\) |
\(\sqrt{a^4} = a^2\) is never negative, so no condition on \(a\) is needed.
Write each one as an entire surd: square the number outside and move it inside.
| \(3\sqrt{7}\) | \(=\) | \(\sqrt{9\times 7} = \sqrt{63}\) |
| \(4\sqrt{5}\) | \(=\) | \(\sqrt{16\times 5} = \sqrt{80}\) |
| \(7\sqrt{2}\) | \(=\) | \(\sqrt{49\times 2} = \sqrt{98}\) |
| \(2\sqrt{15}\) | \(=\) | \(\sqrt{4\times 15} = \sqrt{60}\) |
A larger number under the root gives a larger surd, so the order is:
| \(\) | \(\) | \(\mathbf{2\sqrt{15},\ 3\sqrt{7},\ 4\sqrt{5},\ 7\sqrt{2}}\) |
(i) \(\sqrt[3]{250}\)
(ii) \(\sqrt{4\dfrac{4}{9}}\)
(iii) \(\sqrt{0.18}\)
(i) Look for a cube factor: \(250 = 125\times 2\) and \(125 = 5^3\).
| \(\sqrt[3]{250}\) | \(=\) | \(\sqrt[3]{125}\times\sqrt[3]{2}\) |
| \(\) | \(=\) | \(\mathbf{5\sqrt[3]{2}}\) |
(ii) Change the mixed numeral to an improper fraction, then root the top and bottom.
| \(\sqrt{4\dfrac{4}{9}}\) | \(=\) | \(\sqrt{\dfrac{40}{9}} = \dfrac{\sqrt{40}}{3}\) |
| \(\) | \(=\) | \(\mathbf{\dfrac{2\sqrt{10}}{3}}\) |
(iii) Write the decimal over a square power of ten.
| \(\sqrt{0.18}\) | \(=\) | \(\sqrt{\dfrac{18}{100}} = \dfrac{\sqrt{9\times 2}}{10}\) |
| \(\) | \(=\) | \(\mathbf{\dfrac{3\sqrt{2}}{10}}\) |
(i) the exact length of the hypotenuse, in simplest surd form
(ii) the perimeter, correct to one decimal place.
(i) Use Pythagoras' theorem and leave the answer as a surd.
| \(c^2\) | \(=\) | \(4^2+8^2 = 80\) |
| \(c\) | \(=\) | \(\sqrt{80} = \sqrt{16\times 5}\) |
| \(\) | \(=\) | \(\mathbf{4\sqrt{5}}\text{ cm}\) |
(ii) The exact perimeter is \(12+4\sqrt{5}\) cm. Only now use the calculator:
| \(12+4\sqrt{5}\) | \(=\) | \(20.944\ldots\) |
| \(\) | \(\approx\) | \(\mathbf{20.9}\text{ cm}\) |
\(12+4\sqrt{5}\) is not \(16\sqrt{5}\): a whole number and a surd are unlike terms.
Common pitfalls
Frequently asked questions
How do you simplify a surd?
Write the number under the square root as its largest square factor times another number, then take the square root of the square factor. For example, \(\sqrt{108} = \sqrt{36\times 3} = 6\sqrt{3}\).
What is an entire surd?
An entire surd has no number outside the root, like \(\sqrt{45}\). To change \(3\sqrt{5}\) into an entire surd, square the \(3\) and multiply it in: \(3\sqrt{5} = \sqrt{45}\).
How do you compare surds without a calculator?
Write each one as an entire surd. The surd with the larger number under the root is larger. For example, \(3\sqrt{7} = \sqrt{63}\) and \(2\sqrt{15} = \sqrt{60}\), so \(3\sqrt{7}\) is larger.
Is the square root of a fraction a surd?
It depends. \(\sqrt{\dfrac{16}{25}} = \dfrac{4}{5}\) is rational, but \(\sqrt{\dfrac{40}{9}} = \dfrac{2\sqrt{10}}{3}\) is a surd. Root the top and the bottom separately and simplify.
Why leave an answer as a surd?
A surd is exact, while a decimal from the calculator is rounded. Questions that ask for an exact answer, or for simplest surd form, want the surd. Round only when a decimal answer is asked for, and only at the end.