Lesson 3, Topic 3
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# Square Root of Fractions

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To find the square roots of fractions, reduce fractions to their lowest terms to have perfect squares for both the numerator and denominator.

### Example 1

Find the value of the following

a. $$\sqrt{ \frac{9 }{25}}$$

b. $$\sqrt{ \frac{49 }{4}}$$

c. $$\sqrt{ \frac{18 }{72}}$$

d. $$\sqrt{\scriptsize 2 \normalsize \frac{1 }{4}}$$

e. $$\sqrt{ \frac{72 }{50}}$$

f. $$\sqrt{ \frac{48 }{108}}$$

Solution

a. $$\sqrt{ \frac{9 }{25}} \\ = \frac{\sqrt{9}}{\sqrt{25}}\\ = \frac{3}{5}$$

b. $$\sqrt{ \frac{49 }{4}}\\ = \frac{\sqrt{49}}{\sqrt{4}}\\ = \frac{7}{2}$$

c. $$\sqrt{ \frac{18 }{72}} \\= \frac{\sqrt{2 \: \times \: 9}}{\sqrt{2 \: \times \: 36}} \\ = \frac{\sqrt{9}}{\sqrt{36}}\\= \frac{3}{6} \\ = \frac{1}{2}$$

d. $$\sqrt{\scriptsize 2 \normalsize \frac{1}{4}} \\ = \sqrt{ \frac{9 }{4}} \\= \frac{\sqrt{9}}{\sqrt{4}} \\ = \frac{3}{2} \\ = \scriptsize 1 \normalsize \frac{1}{2}$$

e. $$\sqrt{ \frac{72 }{50}} \\= \frac{\sqrt{2 \: \times \: 36}}{\sqrt{2 \: \times \: 25}}\\ =\sqrt{ \frac{36 }{25}}\\= \frac{6}{5} \\ = \scriptsize 1 \normalsize \frac{1}{5}$$

f. $$\sqrt{ \frac{48}{108}}\\ = \frac{\sqrt{2 \: \times \: 24}}{\sqrt{2 \: \times \: 54}}\\ =\sqrt{ \frac{24 }{54}}\\ =\sqrt{ \frac{12 }{27}}\\ = \frac{\sqrt{4 \: \times \: 3}}{\sqrt{9 \: \times \: 3}}\\ = \sqrt{ \frac{4 }{9}}\\ = \frac{2}{3}$$

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