Simplifying Surds
Simplifying a surd means taking out the largest square factor, so \(\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}\). In NSW Year 9 Advanced maths this is the first step in working with surds (NESA outcome MA5‑IND‑P‑02).
You will simplify surds such as \(\sqrt{252}\) and \(3\sqrt{24}\), simplify \(\dfrac{\sqrt{75}}{15}\) and \(\sqrt{\dfrac{50}{9}}\), and reverse the process to write \(4\sqrt{7}\) as an entire surd. You will compare two surds without a calculator, give the diagonal of a rectangle and the perimeter of a square in exact form, and find the smallest \(k\) that makes \(\sqrt{45k}\) a whole number. It is part of Year 9 Advanced Pythagoras Theorem.
Every question with a fully worked solution.
- Simplifying Surds - Video - Simplifying surds Watch
Theory
To simplify a surd, split off the largest perfect square factor and take its root outside: \(\sqrt{72} = \sqrt{36}\times\sqrt{2} = 6\sqrt{2}\). This Year 9 Advanced maths page also covers entire surds, comparing surds without a calculator, surds in fractions and using prime factors.
A surd is in simplest form when the number under the root has no square factor except \(1\). \(\sqrt{50}\) is not in simplest form, because \(50 = 25 \times 2\). Its simplest form is \(5\sqrt{2}\).
Simplifying uses one rule: \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\). Choose \(a\) to be the largest perfect square that divides the number. Its root comes out as a whole number; the rest stays under the root.
An entire surd has everything under the root, such as \(\sqrt{75}\). A mixed surd has a number in front, such as \(5\sqrt{3}\). The two forms are equal, and changing to an entire surd is the easiest way to compare sizes.
Simplest surd form gives exact answers that are easy to compare and combine. Pythagoras answers, square side lengths and perimeters are often given this way.
For \(a, b \ge 0\), a root can be split across a product or a quotient:
Moving a number into or out of the root:
With pronumerals, \(\sqrt{a^2} = a\) when \(a > 0\):
| Perfect squares to look for |
|---|
| \(4,\ 9,\ 16,\ 25,\ 36,\ 49,\ 64,\ 81,\ 100,\ 121,\ 144,\ 169,\ 196,\ 225\) |
How to simplify a surd
- Find the largest perfect square that divides the number. Work down the list of squares, or use prime factors and pair them up.
- Split the root: \(\sqrt{\text{square} \times \text{rest}} = \sqrt{\text{square}} \times \sqrt{\text{rest}}\).
- Take out the root of the square and multiply it by any number already in front.
- Check the number left under the root. If it still has a square factor, repeat.
- For a fraction, simplify the surd, then cancel common factors of the whole numbers outside the root.
(i) \(\sqrt{288}\)
(ii) \(2\sqrt{45}\).
(i) Look for the largest perfect square that divides \(288\). Try \(144\): \(288 = 144 \times 2\).
| \(\sqrt{288}\) | \(=\) | \(\sqrt{144} \times \sqrt{2}\) |
| \(\) | \(=\) | \(\mathbf{12\sqrt{2}}\) |
A smaller square factor still works, but takes more steps: \(\sqrt{288} = 2\sqrt{72} = 2 \times 6\sqrt{2} = 12\sqrt{2}\). Stop only when the number left under the root has no square factor.
(ii) Simplify the surd, then multiply by the number in front.
| \(2\sqrt{45}\) | \(=\) | \(2 \times \sqrt{9} \times \sqrt{5}\) |
| \(\) | \(=\) | \(2 \times 3 \times \sqrt{5}\) |
| \(\) | \(=\) | \(\mathbf{6\sqrt{5}}\) |
(i) \(\sqrt{\dfrac{27}{16}}\)
(ii) \(\dfrac{\sqrt{108}}{9}\).
(i) Take the root of the top and the bottom separately.
| \(\sqrt{\dfrac{27}{16}}\) | \(=\) | \(\dfrac{\sqrt{27}}{\sqrt{16}}\) |
| \(\) | \(=\) | \(\dfrac{\sqrt{9 \times 3}}{4}\) |
| \(\) | \(=\) | \(\mathbf{\dfrac{3\sqrt{3}}{4}}\) |
(ii) Simplify the surd first, then cancel the whole numbers outside the root.
| \(\dfrac{\sqrt{108}}{9}\) | \(=\) | \(\dfrac{\sqrt{36 \times 3}}{9}\) |
| \(\) | \(=\) | \(\dfrac{6\sqrt{3}}{9}\) |
| \(\) | \(=\) | \(\mathbf{\dfrac{2\sqrt{3}}{3}}\) |
Cancel \(6\) with \(9\), never a number under the root with one outside it.
(i) Write \(5\sqrt{3}\) as an entire surd.
(ii) Without a calculator, decide which is larger: \(2\sqrt{11}\) or \(3\sqrt{5}\).
(i) Write the \(5\) as \(\sqrt{25}\) and combine under one root.
| \(5\sqrt{3}\) | \(=\) | \(\sqrt{25} \times \sqrt{3}\) |
| \(\) | \(=\) | \(\mathbf{\sqrt{75}}\) |
(ii) Write both as entire surds, then compare the numbers under the roots.
| \(2\sqrt{11}\) | \(=\) | \(\sqrt{4 \times 11} = \sqrt{44}\) |
| \(3\sqrt{5}\) | \(=\) | \(\sqrt{9 \times 5} = \sqrt{45}\) |
\(45 > 44\), so \(\boldsymbol{3\sqrt{5}}\) is larger. The bigger number in front, or under the root, does not decide it on its own.
(i) Write \(\sqrt{2^5 \times 5^3}\) in simplest surd form.
(ii) Find the smallest whole number \(k\) for which \(\sqrt{72k}\) is a whole number, and give that whole number.
(i) Split each power into an even power times what is left. An even power is a perfect square: \(\sqrt{2^4} = 2^2\).
| \(2^5 \times 5^3\) | \(=\) | \(2^4 \times 5^2 \times 2 \times 5\) |
| \(\sqrt{2^5 \times 5^3}\) | \(=\) | \(2^2 \times 5 \times \sqrt{10}\) |
| \(\) | \(=\) | \(\mathbf{20\sqrt{10}}\) |
(ii) A perfect square has every prime to an even power. \(72 = 2^3 \times 3^2\). The \(3^2\) is fine, but \(2^3\) needs one more \(2\).
| \(k\) | \(=\) | \(\mathbf{2}\) |
| \(72 \times 2\) | \(=\) | \(2^4 \times 3^2 = 144\) |
| \(\sqrt{144}\) | \(=\) | \(\mathbf{12}\) |
Common pitfalls
Frequently asked questions
How do you simplify a surd?
Find the largest perfect square that divides the number under the root. Split the root into the root of that square times the root of what is left, and take the square's root outside. For example, \(\sqrt{72} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}\).
What is simplest surd form?
A surd is in simplest surd form when the number under the root has no square factor other than \(1\). \(\sqrt{18}\) is not simplest, because \(18\) contains \(9\); \(3\sqrt{2}\) is.
What is an entire surd?
An entire surd has the whole value under the root sign, such as \(\sqrt{75}\). To change \(5\sqrt{3}\) into an entire surd, square the \(5\) and multiply: \(25 \times 3 = 75\).
How do you compare surds without a calculator?
Write each one as an entire surd, then compare the numbers under the roots. The bigger number gives the bigger surd.
Can you simplify root 30?
No. The factors of \(30\) are \(1, 2, 3, 5, 6, 10, 15\) and \(30\), and none except \(1\) is a perfect square, so \(\sqrt{30}\) is already in simplest form.