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Algebra II : Radicals and Fractions

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Example Question #221 : Simplifying Radicals

Rationalize: $\frac{9}{\sqrt{20}}$

Possible Answers:

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

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

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

$\frac{3\sqrt{5}}{10}$

$\frac{9\sqrt{5}}{10}$

Correct answer:

$\frac{9\sqrt{5}}{10}$

Explanation:

We can factor the square root twenty before rationalizing the value so that the final term would not be as difficult to factor.

Factor $\sqrt{20}$ using factors of perfect squares.

$\sqrt{20}= \sqrt{4}\times \sqrt5 = 2\sqrt5$

The expression becomes:

$\frac{9}{\sqrt{20}} = \frac{9}{2\sqrt{5}}$

Instead of multiplying $\sqrt{20}$ on both the top and bottom, we will multiply $\sqrt5$ on the top and bottom.

$\frac{9}{2\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{9\sqrt{5}}{10}$

The answer is: $\frac{9\sqrt{5}}{10}$

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Example Question #222 : Simplifying Radicals

Try without a calculator.

Simplify:

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

Possible Answers:

$\frac{\sqrt{10}}{2}$

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

$\frac{\sqrt{10}}{4}$

None of the other choices gives the correct response.

$\frac{\sqrt{5}}{4}$

Correct answer:

$\frac{\sqrt{10}}{4}$

Explanation:

First, apply the Quotient of Radicals Property to split the radical into a numerator and a denominator:

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

$= \frac{\sqrt{5}}{\sqrt{8}}$

Simplify the denominator by applying the Product of Radicals Property, as follows:

$= \frac{\sqrt{5}}{\sqrt{4} \cdot \sqrt{2}}$

$= \frac{\sqrt{5}}{2\cdot \sqrt{2}}$

Rationalize the denominator by multiplying both halves of the fraction by $\sqrt{2}$ :

$= \frac{\sqrt{5} \cdot \sqrt{2}}{2\cdot \sqrt{2} \cdot \sqrt{2}}$

Apply the Product of Radicals property in the numerator:

$= \frac{\sqrt{5 \cdot 2}}{2\cdot 2}$

$= \frac{\sqrt{10}}{4}$ ,

the correct response.

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Example Question #1551 : Mathematical Relationships And Basic Graphs

Try without a calculator.

Simplify:

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

Possible Answers:

$\frac{2 \sqrt{21} }{7}$

$\frac{ \sqrt{42} }{7}$

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

$\frac{3\sqrt{14} }{7}$

None of these

Correct answer:

$\frac{2 \sqrt{21} }{7}$

Explanation:

First, apply the Quotient of Radicals Property to split the radical into a numerator and a denominator:

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

$= \frac{\sqrt{12}}{\sqrt{7}}$

Simplify the numerator by applying the Product of Radicals Property, as follows:

$\frac{\sqrt{4 \cdot 3 }}{\sqrt{7}}$

$= \frac{\sqrt{4} \cdot \sqrt{3}}{\sqrt{7}}$

$= \frac{2 \sqrt{3}}{\sqrt{7}}$

Rationalize the denominator by multiplying both halves of the fraction by $\sqrt{7}$ :

$\frac{2 \sqrt{3} \cdot \sqrt{7} }{\sqrt{7} \cdot \sqrt{7}}$

Apply the Product of Radicals Property in the numerator:

$\frac{2 \sqrt{3 \cdot 7} }{7}$

$= \frac{2 \sqrt{21} }{7}$

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Algebra II : Radicals and Fractions

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #221 : Simplifying Radicals

Example Question #222 : Simplifying Radicals

Example Question #1551 : Mathematical Relationships And Basic Graphs

All Algebra II Resources

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