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Middle Years

2.03 Multiplying and dividing surds

Worksheet
Multiplication
1

Are the following statements true or false?

a

\sqrt{8^{2}} = \left(\sqrt{8}\right)^{2}

b

\sqrt{5^{2}} = \left(\sqrt{5\times 5}\right)^{2}

c

\sqrt{2^{2}} = \sqrt{2+2}

d

\sqrt{8^{2}} = \sqrt{16}\times \sqrt{4}

2

Complete the following statements by following the example:

\sqrt{ 9 \times 4} = \sqrt{9} \times \sqrt{4}=3 \times 2 = 6
a

\sqrt{ 36 \times 25}=\sqrt{⬚} \times \sqrt{⬚}=⬚ \times ⬚ = ⬚

b

\sqrt{ 9 \times 11} = \sqrt{⬚} \times \sqrt{⬚}=⬚ \sqrt{⬚}

c

\sqrt{ 49 \times 5} = \sqrt{⬚} \times \sqrt{⬚} = ⬚ \sqrt{⬚}

d

\sqrt{ 64 \times 3} = \sqrt{⬚} \times \sqrt{⬚} = ⬚ \sqrt{⬚}

3

Simplify the following:

a
\sqrt{75}
b

\sqrt{19} \times \sqrt{17}

c

\left( 6 \sqrt{8}\right)^{2}

d

(6 \sqrt{3})^2

4

Simplify the following:

a

\sqrt{5} \times \sqrt{7}

b

8 \times 10 \sqrt{5}

c

\sqrt{7} \times \sqrt{3} \times \sqrt{11}

d

\sqrt{55} \times \sqrt{11}

e

4 \sqrt{11} \times 5

f

2 \sqrt{5} \times 15 \sqrt{11}

g

7 \sqrt{22} \times \sqrt{2}

h

\sqrt{180} \times \sqrt{48}

i

8 \sqrt{15} \times 8 \sqrt{5}

j

5 \sqrt{17} \times 8 \sqrt{3}

k

17 \sqrt{35} \times 4 \sqrt{5}

l

8 \sqrt{51} \times 9 \sqrt{3}

5

Simplify the following:

a

\sqrt{11} \left(\sqrt{7} + 4\right)

b

\sqrt{7} \left(3 + \sqrt{3}\right)

c

\sqrt{2} \left(\sqrt{11}-6\right)

d

3 \sqrt{3} \left(\sqrt{13} - 5\right)

e

\sqrt{3} \left(\sqrt{11} + \sqrt{13}\right)

f

4 \sqrt{7} \left(\sqrt{2}-\sqrt{11} \right)

g

3 \sqrt{5} \left(\sqrt{55} + \sqrt{11}\right)

h

8 \sqrt{2} \left(\sqrt{3}- 3 \sqrt{7}\right)

i

5 \sqrt{2} \left( 3 \sqrt{5} + 4 \sqrt{7}\right)

j

7 \sqrt{3} \left( \sqrt{15} + \sqrt{60}\right)

k

11 \sqrt{3} \left( 3 \sqrt{5} - \sqrt{20}\right)

l

8 \sqrt{11} \left( 3 \sqrt{7} - 4 \sqrt{5}\right)

Division
6

Simplify the following:

a

\sqrt{15} \div \sqrt{5}

b

\sqrt{55} \div \sqrt{5}

c

\sqrt{51} \div \sqrt{17}

d

\sqrt{21} \div \sqrt{3}

e

\sqrt{91} \div \sqrt{7}

f

40 \sqrt{7} \div 8

g

10 \sqrt{55} \div \sqrt{11}

h

15 \sqrt{22} \div \sqrt{11}

i

4 \sqrt{35} \div 2 \sqrt{5}

j

\sqrt{27} \div \sqrt{3}

k

3 \sqrt{20} \div \sqrt{5}

l

5 \sqrt{8} \div \sqrt{2}

m

40 \sqrt{96} \div 10 \sqrt{6}

n

50 \sqrt{24} \div 10 \sqrt{6}

o

\sqrt{25} \div \sqrt{81}

p

\sqrt{162} \div \sqrt{8}

7

Simplify the following:

a
\sqrt{\dfrac{28}{7}}
b
\sqrt{\dfrac{9}{45}}
c
\sqrt{\dfrac{64}{4}}
d
\sqrt{\dfrac{48}{144}}
e
\dfrac{\sqrt{12}}{\sqrt{36}}
f
\dfrac{\sqrt{56}}{\sqrt{14}}
g
\dfrac{\sqrt{36}}{\sqrt{81}}
h
\dfrac{\sqrt{72}}{\sqrt{32}}
Applications
8

Find the exact area of the following rectangles.

a
b
9

Find the area of the trapezium in simplified surd form:

10

The body surface area of a person in square metres can be modelled by A = \dfrac{\sqrt{h} \times \sqrt{w}}{60}, where A is the surface area, h is the height of the person in cm, and w is the weight of the person in kg.

a

Use the model to find the surface area of a person who is 164 cm tall and weighs 63 kg. Leave your answer in exact form.

b

Hence find the approximate surface area of the person, to the nearest hundredth of a square metre.

11

Find the exact perpendicular height of a triangle whose area is 40 \sqrt{65} square centimetres and whose base measures 10 \sqrt{13} centimetres.

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