← LearnYear 9 Maths · Algebra — Equations & SurdsFree placement check
Year 9AC9M9N01

Surds

Recognise that the real number system includes the rational numbers and the irrational numbers, and solve problems involving real numbers.

Things you should know.

Method

Multiplying surds
a×b=ab\sqrt{a} \times \sqrt{b} = \sqrt{ab}
Multiply what is under the roots.
Dividing surds
ab=ab\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}
Divide what is under the roots.
Squaring a surd
(a)2=a\left(\sqrt{a}\right)^{2} = a
The root and the square cancel.
Rationalising
1a=aa\frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a}
Never leave a surd on the bottom.

Worked example

e.g. simplify 72\text{simplify } \sqrt{72}
  1. Find the largest perfect square that divides the number36×2=7236 \times 2 = 72
  2. Split the surd into two36×2\sqrt{36} \times \sqrt{2}
  3. Take the root of the perfect square6×26 \times \sqrt{2}
  4. Write the answer626\sqrt{2}

Practice

Simplify the following:

1
50\sqrt{50}
Answer525\sqrt{2}
2
18\sqrt{18}
Answer323\sqrt{2}
3
48\sqrt{48}
Answer434\sqrt{3}
4
75\sqrt{75}
Answer535\sqrt{3}
5
2×8\sqrt{2} \times \sqrt{8}
Answer44
6
3×12\sqrt{3} \times \sqrt{12}
Answer66
7
502\frac{\sqrt{50}}{\sqrt{2}}
Answer55
8
(25)2\left(2\sqrt{5}\right)^{2}
Answer2020
9
32+523\sqrt{2} + 5\sqrt{2}
Answer828\sqrt{2}
10
27−12\sqrt{27} - \sqrt{12}
Answer3\sqrt{3}
11
15\frac{1}{\sqrt{5}}
Answer55\frac{\sqrt{5}}{5}
12
63\frac{6}{\sqrt{3}}
Answer232\sqrt{3}
Next step
Practise Surds with instant marking
A free 10-minute placement check finds which Year 9 topics to work on first, then Summit builds a weekly plan around them.
See where my child is →