Year selective Mathematics | Victorian Curriculum 2.0
Surds & irrational arithmetic
Topic 01 | Content: Year 9 extension | Answer key

Simplify

    1. 232\sqrt{3}.
    2. 333\sqrt{3}.
    3. 353\sqrt{5}.
    4. 727\sqrt{2}.
    5. 232=2×42=822\sqrt{32} = 2 \times 4\sqrt{2} = 8\sqrt{2}.
    6. 10210\sqrt{2}.

Add & subtract

    1. 454\sqrt{5}.
    2. 323\sqrt{2}.
    3. 8+18=22+32=52\sqrt{8} + \sqrt{18} = 2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2}.
    4. 52−42=25\sqrt{2} - 4\sqrt{2} = \sqrt{2}.
    5. 53+33=835\sqrt{3} + 3\sqrt{3} = 8\sqrt{3}.

Multiply

    1. 21\sqrt{21}.
    2. 6106\sqrt{10}.
    3. 66.
    4. 16=4\sqrt{16} = 4.
    5. 23+36=23+62\sqrt{3} + \sqrt{36} = 2\sqrt{3} + 6.

Rationalise

    1. 33\dfrac{\sqrt{3}}{3}.
    2. 1055=25\dfrac{10\sqrt{5}}{5} = 2\sqrt{5}.
    3. 28=28=14=12\dfrac{\sqrt{2}}{\sqrt{8}} = \sqrt{\tfrac{2}{8}} = \sqrt{\tfrac{1}{4}} = \tfrac{1}{2}.

Mixed & apply

    1. side =48=43= \sqrt{48} = 4\sqrt{3} cm.
    2. diagonal =9+25=34= \sqrt{9 + 25} = \sqrt{34}. Already in simplest form.
    3. 18+8−50=32+22−52=0\sqrt{18} + \sqrt{8} - \sqrt{50} = 3\sqrt{2} + 2\sqrt{2} - 5\sqrt{2} = 0.
    4. 2×3×6=2×3×6=36=6\sqrt{2} \times \sqrt{3} \times \sqrt{6} = \sqrt{2 \times 3 \times 6} = \sqrt{36} = 6.

Challenge

    1. (3)2−(2)2=3−2=1(\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1. It’s the difference of squares pattern (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2, used for rationalising binomial surds.
    2. (2+5)2=4+45+5=9+45(2 + \sqrt{5})^2 = 4 + 4\sqrt{5} + 5 = 9 + 4\sqrt{5}.
    3. 12+3×2−32−3=2−34−3=2−3\dfrac{1}{2 + \sqrt{3}} \times \dfrac{2 - \sqrt{3}}{2 - \sqrt{3}} = \dfrac{2 - \sqrt{3}}{4 - 3} = 2 - \sqrt{3}.
    4. Hypotenuse =1+3=2= \sqrt{1 + 3} = 2. Area of square =22=4= 2^2 = 4.
Year selective Mathematics study companion | Answer key