PSAT Math : How to divide square roots

Study concepts, example questions & explanations for PSAT Math

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Example Questions

Example Question #1 : Square Roots And Operations

Divide and simplify. Assume all integers are positive real numbers. 

\(\displaystyle \sqrt{\frac{32}{4}}\)

Possible Answers:

\(\displaystyle \sqrt{2}\)

\(\displaystyle \sqrt{8}\)

\(\displaystyle 4\sqrt{2}\)

\(\displaystyle 2\sqrt{2}\)

Correct answer:

\(\displaystyle 2\sqrt{2}\)

Explanation:

\(\displaystyle \sqrt{\frac{32}{4}}\) 

There are two ways to solve this problem. First you can divide the numbers under the radical. Then simplify. 

Example 1

\(\displaystyle \sqrt{\frac{32}{4}}=\sqrt{8}=\sqrt{4}\cdot\sqrt{2}=2\sqrt{2}\)

Example 2 

Find the square root of both numerator and denominator, simplifying as much as possible then dividing out like terms. 

\(\displaystyle \sqrt{\frac{32}{4}}=\frac{\sqrt{32}}{\sqrt{4}}=\frac{\sqrt{16}\cdot \sqrt{2}}{\sqrt{4}}=\frac{4\sqrt{2}}{2}=2\sqrt{2}\)

Both methods will give you the correct answer of \(\displaystyle 2\sqrt{2}\).

Example Question #1 : How To Divide Square Roots

(√27 + √12) / √3 is equal to

Possible Answers:

18

5/√3

5

√3

(6√3)/√3

Correct answer:

5

Explanation:

√27 is the same as 3√3, while √12 is the same as 2√3.

3√3 + 2√3 = 5√3

(5√3)/(√3) = 5

Example Question #3 : Basic Squaring / Square Roots

Simplify:

\(\displaystyle \sqrt{\frac{64}{169}}\)

Possible Answers:

\(\displaystyle \frac{8}{13}\)

\(\displaystyle \frac{8}{15}\)

\(\displaystyle \frac{64}{169}\)

\(\displaystyle \frac{7}{13}\)

Correct answer:

\(\displaystyle \frac{8}{13}\)

Explanation:

To simplfy, we must first distribute the square root.

\(\displaystyle \frac{\sqrt{64}}{\sqrt{169}}\)

Next, we can simplify each of the square roots.

\(\displaystyle \frac{8}{13}\)

Example Question #3 : Basic Squaring / Square Roots

Find the quotient:

\(\displaystyle \sqrt{\frac{18}{2}}\)

Possible Answers:

\(\displaystyle 9\)

\(\displaystyle \frac{\sqrt{2}}{3}\)

\(\displaystyle 3\)

\(\displaystyle \frac{3\sqrt{2}}{2}\)

\(\displaystyle \sqrt{2}\)

Correct answer:

\(\displaystyle 3\)

Explanation:

Find the quotient:

\(\displaystyle \frac{\sqrt{18}}{\sqrt{2}}\)

There are two ways to approach this problem.

Option 1: Combine the radicals first, the reduce

\(\displaystyle \frac{\sqrt{18}}{\sqrt{2}}=\sqrt{\frac{18}{2}}=\sqrt{9}=3\)

Option 2: Simplify the radicals first, then reduce

\(\displaystyle \frac{\sqrt{18}}{\sqrt{2}}=\frac{\sqrt{9\cdot 2}}{\sqrt{2}}=\frac{3\sqrt{2}}{\sqrt{2}}=3\)

Example Question #1 : How To Divide Square Roots

Find the quotient:

\(\displaystyle \frac{\sqrt{54}}{\sqrt{45}}\)

Possible Answers:

\(\displaystyle 3\sqrt{6}\)

\(\displaystyle 9\)

\(\displaystyle 9\sqrt{5}\)

\(\displaystyle \frac{\sqrt{}6}{\sqrt{}5}\)

\(\displaystyle \frac{\sqrt{30}}{5}\)

Correct answer:

\(\displaystyle \frac{\sqrt{30}}{5}\)

Explanation:

 

Simplify each radical:

\(\displaystyle \frac{\sqrt{54}}{\sqrt{45}} = \frac{\sqrt{9\cdot 6}}{\sqrt{9\cdot 5}}=\frac{3\sqrt{6}}{3\sqrt{5}}=\frac{\sqrt{6}}{\sqrt{5}}\)

 

Rationalize the denominator:

\(\displaystyle \frac{\sqrt{6}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}= \frac{\sqrt{30}}{5}\)

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