Lesson 3, Topic 1
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# Multiplication & Division of Square Roots

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### 1. For equations involving multiplication

i. Split the square roots if both numbers are perfect squares e.g.

$$\scriptsize \sqrt{4 \: \times \: 16} \\ \scriptsize = \sqrt{4} \: \times \: \sqrt{16} \\ \scriptsize = 2 \: \times \: 4 \\ \scriptsize = 8$$

ii.  Multiply directly, if one or both numbers are not perfect squares e.g.

$$\scriptsize \sqrt{2 \: \times \: 50}\\ \scriptsize = \sqrt{100}\\ \scriptsize = 10$$

This is because 2 and 50 are not perfect squares, but their product 100 is.

### 2. For equations involving Division

i. Split the square roots, if both numbers are perfect squares e.g.

$$\sqrt{ \frac{144 }{81}}\\= \frac{\sqrt{144}}{\sqrt{81}} \\ = \frac{12}{9} \\ \ = \frac{4}{3} \\ \scriptsize = 1 \normalsize \frac{1}{3}$$

ii. Divide directly if one or both numbers are not perfect squares e.g.

$$\scriptsize \sqrt{128 \: \div \: 2}\\ \scriptsize =\sqrt{64} \\ \scriptsize = 8$$

Example 1

Evaluate the following;

a. $$\scriptsize \sqrt{24 + 1}$$

b. $$\scriptsize \sqrt{50 \: – \: 25}$$

c. $$\scriptsize \sqrt{200 \: – \: 4}$$

d. $$\scriptsize \sqrt{10 + 20 + 30 + 50 \: – \: 10}$$

e. $$\scriptsize \sqrt{64 \: \times \: 81}$$

f. $$\sqrt{ \frac{400}{4}}$$

g. $$\scriptsize \sqrt{4 \: \times \: 6 \: \times \: 6}$$

h. $$\scriptsize \sqrt{ 9 \: \times \: 8 \: \div \: 2}$$

i. $$\scriptsize \sqrt{ 27 \: \times \: 3 \: \div \: 9}$$

Solution

a. $$\scriptsize \sqrt{24 \:+\:1} \\ \scriptsize = \sqrt{25}\\ \scriptsize = 5$$

b. $$\scriptsize \sqrt{50 \: – \: 25} \\ \scriptsize = \sqrt{25}\\ \scriptsize = 5$$

c. $$\scriptsize \sqrt{200 \: – \: 4} \\ \scriptsize = \sqrt{196} \\ \scriptsize = 14$$

d. $$\scriptsize \sqrt{10\: + \: 20\: +\: 30 \:+ \: 50 \: – \: 10}\\ \scriptsize =\sqrt{110\: – \:10}\\ \scriptsize = \sqrt{100} \\ \scriptsize= 10$$

e. $$\scriptsize \sqrt{64 \: \times \: 81} \\ \scriptsize = \sqrt{64} \: \times \sqrt{81} \\ \scriptsize = 8 \: \times \: 9 \\ \scriptsize = 72$$

f. $$\sqrt{ \frac{400}{4}}\\ \scriptsize = \sqrt{100} \\ \scriptsize = 10$$

or

$$\sqrt{ \frac{400}{4}}\\ = \frac{\sqrt{400}}{\sqrt{4}}\\ = \frac{20}{2} \\ \scriptsize = 10$$

g. $$\scriptsize \sqrt{4 \: \times \: 6 \: \times \: 6}$$

= $$\scriptsize \sqrt{4 \: \times \: 36}\\ \scriptsize = \sqrt{4} \: \times \: \sqrt{36}$$

= $$\scriptsize 2 \: \times \: 6 = 12$$

h. $$\scriptsize \sqrt{ 9 \: \times \: 8 \: \div \: 2}\\ \scriptsize = \sqrt{9 \: \times \: 4}$$

= $$\scriptsize \sqrt{9} \: \times \: \sqrt{4}$$

= $$\scriptsize 3 \: \times \: 2 = 6$$

i. In this case apply BODMAS (Do the division first)

$$\scriptsize \sqrt{ 27 \: \times \: 3 \: \div \: 9}\\ \scriptsize = \sqrt{ \normalsize \frac{27 \: \times \: 3 }{9}} \\ \scriptsize = \sqrt{3 \: \times \: 3}$$

$$\scriptsize = \sqrt{9} = 3$$
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