How to divide square roots

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SSAT Upper Level Quantitative › How to divide square roots

Questions 1 - 8

1

Simplify by rationalizing the denominator:

$\frac{6}{\sqrt{5}+\sqrt{7}}$

The correct answer is not among the other responses.

$\frac{ \sqrt{7}+ \sqrt{5}}{12}$

$\frac{ \sqrt{7}- \sqrt{5}}{12}$

$\frac{ \sqrt{7}+ \sqrt{5}}{3}$

$\frac{ \sqrt{7}- \sqrt{5}}{3}$

Explanation

Multiply both numerator and denominator by the conjugate of the denominator, which is $\sqrt{5}-\sqrt{7}$ :

$\frac{6}{\sqrt{5}+\sqrt{7}}$

$=\frac{6(\sqrt{5}-\sqrt{7})}{(\sqrt{5}+\sqrt{7})(\sqrt{5}-\sqrt{7})}$

$=\frac{6(\sqrt{5}-\sqrt{7})}{(\sqrt{5})^{2}-(\sqrt{7})^{2}}$

$=\frac{6(\sqrt{5}-\sqrt{7})}{(\sqrt{5})^{2}-(\sqrt{7})^{2}}$

$=\frac{6(\sqrt{5}-\sqrt{7})}{5-7}$

$=\frac{6(\sqrt{5}-\sqrt{7})}{-2}$

$= -3(\sqrt{5}-\sqrt{7} )$

$= 3\sqrt{7}-3\sqrt{5}$

This is not among the given choices.

2

Simplify by rationalizing the denominator:

$\frac{16}{4-\sqrt{6}}$

$\frac{ 32+8\sqrt{6}}{5}$

$\frac{ 32-8\sqrt{6}}{5}$

$- 32-8\sqrt{6}$

$\frac{ 32+8\sqrt{6}}{11}$

$\frac{ 32-8\sqrt{6}}{11}$

Explanation

Multiply both numerator and denominator by the conjugate of the denominator, which is $4+\sqrt{6}$ :

$\frac{16}{4-\sqrt{6}}= \frac{16(4+\sqrt{6})}{(4-\sqrt{6}) (4+\sqrt{6})}$

$= \frac{16(4+\sqrt{6})}{4^{2}-(\sqrt{6})^{2} }$

$= \frac{16(4+\sqrt{6})}{16-6}$

$= \frac{16(4+\sqrt{6})}{10}$

$= \frac{8(4+\sqrt{6})}{5}$

$= \frac{ 32+8\sqrt{6}}{5}$

3

Divide:

$18\textrm{ lbs }2\textrm{ oz } \div 5$

$3\textrm{ lbs }10 \textrm{ oz }$

$3\textrm{ lbs }8 \textrm{ oz }$

$3\textrm{ lbs }6 \textrm{ oz }$

$3\textrm{ lbs }10 \textrm{ oz }$

$3\textrm{ lbs }12 \textrm{ oz }$

Explanation

We can write 18 pounds, 2 ounces as just ounces as follows:

$18\textrm{ lbs }2\textrm{ oz } = (18 \times 16 + 2) \textrm{ oz } = 290 \textrm{ oz }$

Divide:

$290 \div 5 = 58$

Convert 58 ounces to pounds and ounces as follows:

$58 \div 16 = 3 \textrm{ R }10$

That is, 3 pounds 10 ounces.

4

Simplify by rationalizing the denominator:

$\frac{\sqrt{15}}{\sqrt{5}-\sqrt{3}}$

$\frac{ 5\sqrt {3}+3 \sqrt{5}}{2 }$

$5\sqrt {3}+3 \sqrt{5}$

$\frac{ 5\sqrt {3}-3 \sqrt{5}}{2 }$

$\frac{ 5\sqrt {3}+3 \sqrt{5}}{8 }$

$\frac{ 5\sqrt {3}-3 \sqrt{5}}{8 }$

Explanation

Multiply both numerator and denominator by the conjugate of the denominator, which is $\sqrt{5}+\sqrt{3}$

$\frac{\sqrt{15}}{\sqrt{5}-\sqrt{3}}$

$=\frac{\sqrt{15}( \sqrt{5}+\sqrt{3})}{\left (\sqrt{5}-\sqrt{3} \right )( \sqrt{5}+\sqrt{3})}$

$=\frac{ \sqrt{15} \cdot \sqrt{5}+ \sqrt{15} \cdot \sqrt{3}}{ (\sqrt{5})^{2}-(\sqrt{3} )^{2} }$

$=\frac{ \sqrt{15 \cdot 5}+ \sqrt{15 \cdot 3}}{5-3 }$

$=\frac{ \sqrt{75}+ \sqrt{45}}{2 }$

$=\frac{ \sqrt{25} \cdot \sqrt {3}+ \sqrt{9} \cdot \sqrt{5}}{2 }$

$=\frac{ 5\sqrt {3}+3 \sqrt{5}}{2 }$

5

Simplify by rationalizing the denominator:

$\frac{\sqrt{6}}{6-\sqrt{3}}$

$\frac{\sqrt{2}+ 2 \sqrt{6}}{11 }$

$\frac{-\sqrt{2}+ 2 \sqrt{6}}{11 }$

$\frac{12-2\sqrt{3}}{13}$

$\frac{-\sqrt{2}+ 2 \sqrt{6}}{13}$

$\frac{12+2\sqrt{3}}{11}$

Explanation

Multiply both numerator and denominator by the conjugate of the denominator, which is $6+\sqrt{3}$ :

$\frac{\sqrt{6}}{6-\sqrt{3}}$

$= \frac{\sqrt{6}(6+\sqrt{3})}{(6-\sqrt{3})(6+\sqrt{3})}$

$= \frac{6\cdot \sqrt{6}+\sqrt{3} \cdot \sqrt{6}}{6^{2}-(\sqrt{3})^{2}}$

$= \frac{6 \sqrt{6}+ \sqrt{18}}{36-3 }$

$= \frac{6 \sqrt{6}+ \sqrt{9} \cdot \sqrt{2}}{3 3 }$

$= \frac{6 \sqrt{6}+3\sqrt{2}}{3 3 }$

$= \frac{2 \sqrt{6}+\sqrt{2}}{11 }$

$= \frac{\sqrt{2}+ 2 \sqrt{6}}{11 }$

6

Simplify by rationalizing the denominator:

$\frac{3\sqrt{2}+2\sqrt{3}}{\sqrt{6}}$

$\sqrt{3}+ \sqrt{2}$

$\frac{9\sqrt{3}+4 \sqrt{2} }{6}$

$\frac{\sqrt{3}+ \sqrt{2}}{2}$

$\frac{9\sqrt{3}+4 \sqrt{2} }{4}$

$\frac{9\sqrt{3}+4 \sqrt{2} }{9}$

Explanation

Multiply both numerator and denominator by $\sqrt{6}$ :

$\frac{3\sqrt{2}+2\sqrt{3}}{\sqrt{6}}$

$= \frac{(3\sqrt{2}+2\sqrt{3})\cdot \sqrt{6}}{\sqrt{6}\cdot \sqrt{6}}$

$= \frac{3\sqrt{2}\cdot \sqrt{6}+2\sqrt{3}\cdot \sqrt{6}}{6}$

$= \frac{3\sqrt{12} +2\sqrt{18} }{6}$

$= \frac{3\cdot \sqrt{4} \cdot \sqrt{3}+2\cdot \sqrt{9}\cdot \sqrt{2} }{6}$

$= \frac{3\cdot 2 \cdot \sqrt{3}+2\cdot3 \cdot \sqrt{2} }{6}$

$= \frac{6 \cdot \sqrt{3}+6 \cdot \sqrt{2} }{6}$

$= \sqrt{3}+ \sqrt{2}$

7

Simplify by rationalizing the denominator:

$\frac{12}{\sqrt{14}-\sqrt{5}}$

$\frac{4 \sqrt{14}+4\sqrt{5}}{3}$

$\frac{4 \sqrt{14}-4\sqrt{5}}{3}$

$\frac{12 \sqrt{14}-12\sqrt{5}}{19}$

$\frac{12 \sqrt{14}+12\sqrt{5}}{19}$

None of the other responses is correct.

Explanation

Multiply both numerator and denominator by the conjugate of the denominator, which is $\sqrt{14}+\sqrt{5}$ :

$\frac{12}{\sqrt{14}-\sqrt{5}}$

$= \frac{12(\sqrt{14}+\sqrt{5})}{(\sqrt{14}-\sqrt{5})(\sqrt{14}+\sqrt{5})}$

$= \frac{12(\sqrt{14}+\sqrt{5})}{(\sqrt{14})^{2}-(\sqrt{5})^{2} }$

$= \frac{12(\sqrt{14}+\sqrt{5})}{14-5}$

$= \frac{12(\sqrt{14}+\sqrt{5})}{9}$

$= \frac{4(\sqrt{14}+\sqrt{5})}{3}$

$= \frac{4 \sqrt{14}+4\sqrt{5}}{3}$

8

Divide: $\frac{\sqrt{64}}{\sqrt{8}}$

$2\sqrt2$

$\frac{1}{8}$

$\frac{\sqrt2}{4}$

Explanation

The values are of the same power. Divide the terms.

$\frac{\sqrt{64}}{\sqrt{8}}= \sqrt{\frac{64}{8}}= \sqrt{8} = \sqrt{4}\times \sqrt{2}= 2\sqrt2$

The correct answer is: $2\sqrt2$

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