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Algebra II : Radicals and Fractions

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Example Questions

Example Question #201 : Simplifying Radicals

Rationalize the denominator: $\frac{6}{\sqrt{28}}$

Possible Answers:

$\frac{7\sqrt3}{3}$

$\frac{3\sqrt7}{7}$

$\sqrt7$

$\frac{3\sqrt5}{4}$

$\textup{The answer is not given.}$

Correct answer:

$\frac{3\sqrt7}{7}$

Explanation:

In order to rationalize the denominator, multiply both the top and bottom by square root of 28.

$\frac{6}{\sqrt{28}}\cdot \frac{\sqrt{28}}{\sqrt{28}}$

Square root 28 can also be written as:

$\sqrt{(4\cdot 7)} = \sqrt{4} \cdot \sqrt{7} = 2\sqrt7$

This means that:

$\frac{6}{\sqrt{28}}\cdot \frac{\sqrt{28}}{\sqrt{28}}=\frac{6}{ 2\sqrt7}\cdot \frac{ 2\sqrt7}{ 2\sqrt7}$

Simplify the terms.

$\frac{6}{ 2\sqrt7}\cdot \frac{ 2\sqrt7}{ 2\sqrt7} = \frac{12\sqrt7}{4\times 7} = \frac{3\sqrt7}{7}$

The answer is: $\frac{3\sqrt7}{7}$

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Example Question #1533 : Mathematical Relationships And Basic Graphs

Rationalize the denominator, if possible: $\frac{\sqrt{3}}{\sqrt{26}}$

Possible Answers:

$\frac{5\sqrt{11}}{3}$

$\textup{The fraction is already simplified.}$

$\frac{\sqrt{78}}{26}$

$\frac{2\sqrt{13}}{13}$

$\frac{19\sqrt{3}}{26}$

Correct answer:

$\frac{\sqrt{78}}{26}$

Explanation:

Multiply the numerator and denominator with denominator. This will eliminate the radical in the denominator.

$\frac{\sqrt{3}}{\sqrt{26}}\cdot\frac{\sqrt{26}}{\sqrt{26}} = \frac{\sqrt{3}\cdot \sqrt{26}}{26} = \frac{\sqrt{78}}{26}$

There are no perfect squares that can be used to factor $\sqrt{78}$ .

The answer is: $\frac{\sqrt{78}}{26}$

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Example Question #1534 : Mathematical Relationships And Basic Graphs

Rationalize the denominator: $\frac{\sqrt{5}}{\sqrt{60}}$

Possible Answers:

$\frac{\sqrt{3}}{9}$

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

$\frac{5\sqrt{3}}{6}$

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

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

Correct answer:

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

Explanation:

To rationalize the denominator, multiply the top and bottom of the fraction by the denominator.

$\frac{\sqrt{5}}{\sqrt{60}}\cdot \frac{\sqrt{60}}{\sqrt{60}}$

Simplify the numerator and denominator.

$\frac{\sqrt{300}}{60}$

Rewrite the numerator based on common factors of known perfect squares.

$\frac{\sqrt{300}}{60}=\frac{\sqrt{25}\cdot \sqrt{12}}{60} = \frac{\sqrt{25}\cdot \sqrt{4} \cdot \sqrt{3}}{60}$

Simplify the numerator.

$\frac{5\times 2 \times \sqrt{3}}{60} = \frac{10\sqrt{3}}{60} = \frac{\sqrt{3}}{6}$

The answer is: $\frac{\sqrt{3}}{6}$

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Example Question #1531 : Mathematical Relationships And Basic Graphs

Rationalize the denominator: $\frac{3\sqrt{3}}{\sqrt{15}}$

Possible Answers:

$5\sqrt3$

$\frac{3\sqrt5}{5}$

$\frac{5\sqrt3}{3}$

$\frac{2\sqrt3}{3}$

$\frac{5\sqrt5}{3}$

Correct answer:

$\frac{3\sqrt5}{5}$

Explanation:

Multiply the top and bottom of the fraction by the denominator.

$\frac{3\sqrt{3}}{\sqrt{15}}\cdot \frac{\sqrt{15}}{\sqrt{15}} = \frac{3\sqrt{45}}{15}$

The radical can be factored using perfect squares.

$\frac{3\sqrt{45}}{15} = \frac{3\cdot \sqrt{9}\cdot \sqrt{5}}{15}= \frac{3\cdot 3\cdot \sqrt{5}}{15}$

Reduce the fraction.

The answer is: $\frac{3\sqrt5}{5}$

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Example Question #31 : Radicals And Fractions

Simplify the radical: $\frac{1}{\sqrt2}+\frac{1}{\sqrt3}$

Possible Answers:

$\frac{\sqrt{10}}{6}$

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

$\frac{\sqrt{10}+2\sqrt{3}}{6}$

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

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

Correct answer:

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

Explanation:

Determine the least common denominator by multiplying both denominators together.

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

Convert both fractions.

$\frac{1}{\sqrt2}+\frac{1}{\sqrt3}=\frac{\sqrt3}{\sqrt2 \cdot \sqrt3}+\frac{\sqrt2}{\sqrt3 \cdot \sqrt2 } = \frac{\sqrt{3}+\sqrt2}{\sqrt6}$

Rationalize the denominator by multiplying square root six on the numerator and the denominator.

$\frac{\sqrt{3}+\sqrt2}{\sqrt6} \cdot \frac{\sqrt6}{\sqrt{6}} = \frac{\sqrt{18}+\sqrt{12}}{6}$

Factor the two radicals on the numerator by perfect squares.

$\sqrt{18} = \sqrt{9}\cdot \sqrt{2} = 3\sqrt2$

$\sqrt{12} = \sqrt{4}\cdot \sqrt{3} = 2\sqrt3$

Replace the terms.

The answer is: $\frac{3\sqrt{2}+2\sqrt{3}}{6}$

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Example Question #31 : Radicals And Fractions

Simplify: $\frac{30}{\sqrt{3}}$

Possible Answers:

$3\sqrt3$

$9\sqrt{30}$

$10\sqrt3$

$\frac{3\sqrt3}{10}$

$\frac{\sqrt{13}}{30}$

Correct answer:

$10\sqrt3$

Explanation:

Multiply both the top and bottom by the denominator.

$\frac{30}{\sqrt{3}}\cdot \frac{\sqrt3}{\sqrt3}$

Simplify both the top and bottom, and reduce the fraction.

$\frac{30\sqrt3}{3} = 10\sqrt3$

The answer is: $10\sqrt3$

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Example Question #1538 : Mathematical Relationships And Basic Graphs

Simplify the fraction: $\frac{21}{\sqrt{7}}$

Possible Answers:

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

$3\sqrt{7}$

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

$\frac{\sqrt{21}}{3}$

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

Correct answer:

$3\sqrt{7}$

Explanation:

Multiply the numerator and denominator by the denominator.

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

Reduce the fraction.

The answer is: $3\sqrt{7}$

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Example Question #1533 : Mathematical Relationships And Basic Graphs

Simplify: $\frac{\sqrt[4]{32}}{4}$

Possible Answers:

$\sqrt[4]{2}$

$\frac{\sqrt[4]{2}}{2}$

$\textup{The expression is simplified as is.}$

$\frac{\sqrt[4]{2}}{4}$

$\frac{\sqrt[4]{2}}{8}$

Correct answer:

$\frac{\sqrt[4]{2}}{2}$

Explanation:

In order to simplify this expression, rewrite the inner term of the fourth root using factors of the power of four, such as:

1^4=1, 2^4=16, 3^4 = 81

Rewrite the fourth root and simplify.

$\frac{\sqrt[4]{32}}{4} = \frac{\sqrt[4]{16 \cdot2}}{4} = \frac{\sqrt[4]{16} \cdot\sqrt[4]{2}}{4} = \frac{2\sqrt[4]{2}}{4}$

Simplify this fraction.

The answer is: $\frac{\sqrt[4]{2}}{2}$

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Example Question #1540 : Mathematical Relationships And Basic Graphs

Rationalize the denominator: $\frac{2}{2+\sqrt{3}}$

Possible Answers:

$2-3\sqrt3$

$2-4\sqrt3$

$4-2\sqrt3$

$2-4\sqrt3$

Correct answer:

$4-2\sqrt3$

Explanation:

Multiply the top and bottom by the conjugate of the denominator.

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

Simplify the top and bottom of the fractions.

$\frac{2}{2+\sqrt{3}}\cdot \frac{2-\sqrt{3}}{2-\sqrt{3}} =\frac{4-2\sqrt3}{4-3}=4-2\sqrt3$

The answer is: $4-2\sqrt3$

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Example Question #301 : Radicals

Solve the radicals: $\frac{2}{\sqrt{3(2+3)}}$

Possible Answers:

$\frac{2\sqrt{30}}{15}$

$\frac{2\sqrt{15}}{15}$

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

$\frac{\sqrt{30}}{15}$

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

Correct answer:

$\frac{2\sqrt{15}}{15}$

Explanation:

Simplify the parentheses first.

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

Rationalize the denominator by multiplying both the top and the bottom by square root 15.

$\frac{2}{\sqrt{15}}\cdot \frac{\sqrt{15}}{\sqrt{15}} = \frac{2\sqrt{15}}{15}$

The answer is: $\frac{2\sqrt{15}}{15}$

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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 #201 : Simplifying Radicals

Example Question #1533 : Mathematical Relationships And Basic Graphs

Example Question #1534 : Mathematical Relationships And Basic Graphs

Example Question #1531 : Mathematical Relationships And Basic Graphs

Example Question #31 : Radicals And Fractions

Example Question #31 : Radicals And Fractions

Example Question #1538 : Mathematical Relationships And Basic Graphs

Example Question #1533 : Mathematical Relationships And Basic Graphs

Example Question #1540 : Mathematical Relationships And Basic Graphs

Example Question #301 : Radicals

All Algebra II Resources

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