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PSAT Math : How to divide square roots

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Example Questions

Example Question #1 : Square Roots And Operations

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

$\sqrt{\frac{32}{4}}$ $\displaystyle \sqrt{\frac{32}{4}}$

Possible Answers:

$\sqrt{2}$ $\displaystyle \sqrt{2}$

$\sqrt{8}$ $\displaystyle \sqrt{8}$

$4\sqrt{2}$ $\displaystyle 4\sqrt{2}$

$2\sqrt{2}$ $\displaystyle 2\sqrt{2}$

Correct answer:

$2\sqrt{2}$ $\displaystyle 2\sqrt{2}$

Explanation:

$\sqrt{\frac{32}{4}}$ $\displaystyle \sqrt{\frac{32}{4}}$

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

Example 1

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

Example 2

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

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

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

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Example Question #1 : How To Divide Square Roots

(√27 + √12) / √3 is equal to

Possible Answers:

5/√3

√3

(6√3)/√3

Correct answer:

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

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Example Question #3 : Basic Squaring / Square Roots

Simplify:

$\sqrt{\frac{64}{169}}$ $\displaystyle \sqrt{\frac{64}{169}}$

Possible Answers:

$\frac{8}{13}$ $\displaystyle \frac{8}{13}$

$\frac{8}{15}$ $\displaystyle \frac{8}{15}$

$\frac{64}{169}$ $\displaystyle \frac{64}{169}$

$\frac{7}{13}$ $\displaystyle \frac{7}{13}$

Correct answer:

$\frac{8}{13}$ $\displaystyle \frac{8}{13}$

Explanation:

To simplfy, we must first distribute the square root.

$\frac{\sqrt{64}}{\sqrt{169}}$ $\displaystyle \frac{\sqrt{64}}{\sqrt{169}}$

Next, we can simplify each of the square roots.

$\frac{8}{13}$ $\displaystyle \frac{8}{13}$

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Example Question #3 : Basic Squaring / Square Roots

Find the quotient:

$\sqrt{\frac{18}{2}}$ $\displaystyle \sqrt{\frac{18}{2}}$

Possible Answers:

$\displaystyle 9$

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

$\displaystyle 3$

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

$\sqrt{2}$ $\displaystyle \sqrt{2}$

Correct answer:

$\displaystyle 3$

Explanation:

Find the quotient:

$\frac{\sqrt{18}}{\sqrt{2}}$ $\displaystyle \frac{\sqrt{18}}{\sqrt{2}}$

There are two ways to approach this problem.

Option 1: Combine the radicals first, the reduce

$\frac{\sqrt{18}}{\sqrt{2}}=\sqrt{\frac{18}{2}}=\sqrt{9}=3$ $\displaystyle \frac{\sqrt{18}}{\sqrt{2}}=\sqrt{\frac{18}{2}}=\sqrt{9}=3$

Option 2: Simplify the radicals first, then reduce

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

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Example Question #1 : How To Divide Square Roots

Find the quotient:

$\frac{\sqrt{54}}{\sqrt{45}}$ $\displaystyle \frac{\sqrt{54}}{\sqrt{45}}$

Possible Answers:

$3\sqrt{6}$ $\displaystyle 3\sqrt{6}$

$\displaystyle 9$

$9\sqrt{5}$ $\displaystyle 9\sqrt{5}$

$\frac{\sqrt{}6}{\sqrt{}5}$ $\displaystyle \frac{\sqrt{}6}{\sqrt{}5}$

$\frac{\sqrt{30}}{5}$ $\displaystyle \frac{\sqrt{30}}{5}$

Correct answer:

$\frac{\sqrt{30}}{5}$ $\displaystyle \frac{\sqrt{30}}{5}$

Explanation:

Simplify each radical:

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

Rationalize the denominator:

$\frac{\sqrt{6}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}= \frac{\sqrt{30}}{5}$ $\displaystyle \frac{\sqrt{6}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}= \frac{\sqrt{30}}{5}$

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PSAT Math : How to divide square roots

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #1 : Square Roots And Operations

Example Question #1 : How To Divide Square Roots

Example Question #3 : Basic Squaring / Square Roots

Example Question #3 : Basic Squaring / Square Roots

Example Question #1 : How To Divide Square Roots

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