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Algebra II : Simplifying Radicals

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #25 : Multiplying And Dividing Radicals

Simplify.

$\sqrt{6}\cdot3\sqrt{6}$

Possible Answers:

$4\sqrt{6}$

$6\sqrt{3}$

Correct answer:

Explanation:

$\sqrt{6}\cdot3\sqrt{6}$ When multiplying radicals, you can combine them and multiply the numbers inside the radical. NUmbers outside the radical are also multiplied.

$3\sqrt{36}$ We can simplify this.

$3\cdot6=18$

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Example Question #25 : Multiplying And Dividing Radicals

Simplify.

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

Possible Answers:

$\sqrt{18}$

$3\sqrt{2}$

$9\sqrt{2}$

$2\sqrt{3}$

Correct answer:

$3\sqrt{2}$

Explanation:

$\sqrt{6}\cdot\sqrt{3}$ When multiplying radicals, you can combine them and multiply the numbers inside the radical.

$\sqrt{18}$ We can simplify this. Lets find a perfect square.

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

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Example Question #29 : Multiplying And Dividing Radicals

Simplify.

$\frac{\sqrt{24}}{4}$

Possible Answers:

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

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

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

$\frac{\sqrt{24}}4{}$

Correct answer:

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

Explanation:

When dividing radicals with an integer, let's simplify the radical. We need to find a perfect square.

$\frac{\sqrt{24}}{4}=\frac{\sqrt{4}\cdot\sqrt{6}}{4}=\frac{2\sqrt{6}}{4}$ . We can reduce.

$\frac{2\sqrt{6}}{4}=\frac{\sqrt{6}}{2}$

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Example Question #30 : Multiplying And Dividing Radicals

Simplify.

$\frac{\sqrt{48}}{16}$

Possible Answers:

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

$4\sqrt{3}$

$2\sqrt{3}$

$\sqrt{3}$

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

Correct answer:

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

Explanation:

When dividing radicals with an integer, let's simplify the radical. We need to find a perfect square.

$\frac{\sqrt{48}}{16}=\frac{\sqrt{16}\cdot\sqrt{3}}{16}=\frac{4\sqrt{3}}{16}$ . We can reduce.

$\frac{4\sqrt{3}}{16}=\frac{\sqrt{3}}{4}$

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Example Question #131 : Simplifying Radicals

Simplify.

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

Possible Answers:

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

$\frac{\sqrt{35}}{5}$

$\frac{7}{5}$

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

$\frac{5}{7}$

Correct answer:

$\frac{\sqrt{35}}{5}$

Explanation:

When dividing radicals, we multiply top and bottom by the bottom radical to ensure our denominator is an integer.

$\frac{\sqrt{7}}{\sqrt{5}}=\frac{\sqrt{7}\cdot\sqrt{5}}{\sqrt{5}\cdot\sqrt{5}}=\frac{\sqrt{35}}{5}$ . This can't be simplified and is our answer.

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Example Question #132 : Simplifying Radicals

Simplify.

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

Possible Answers:

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

$3\sqrt{3}$

$\sqrt{3}$

Correct answer:

$\sqrt{3}$

Explanation:

When dividing radicals, we multiply top and bottom by the bottom radical to ensure our denominator is an integer.

$\frac{3}{\sqrt{3}}=\frac{3\cdot\sqrt{3}}{\sqrt{3}\cdot\sqrt{3}}=\frac{3\sqrt{3}}{3}=\sqrt{3}$ .

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Example Question #33 : Multiplying And Dividing Radicals

Simplify.

$\frac{2}{1+\sqrt{3}}$

Possible Answers:

$-1+\sqrt{3}$

$1-\sqrt{3}$

$-2+2\sqrt{3}$

$\frac{2}{1-\sqrt{3}}$

$\frac{1-\sqrt{3}}{2}$

Correct answer:

$-1+\sqrt{3}$

Explanation:

$\frac{2}{1+\sqrt{3}}$ To get rid of the radical in the denominator, we need to multiply top and bottom by the conjugate which is the opposite sign of the radical expression. This would be $1-\sqrt{3}$ .

$\frac{2}{1+\sqrt{3}}\cdot\frac{1-\sqrt{3}}{1-\sqrt{3}}=\frac{2-2\sqrt{3}}{1-3}=\frac{2-2\sqrt{3}}{-2}$ Remember to FOIL out when multiplying out the denominators. Now, with our answer, we can factor out a .

$\frac{2-2\sqrt{3}}{-2}=\frac{-2(-1+\sqrt{3})}{-2}=-1+\sqrt{3}$

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Example Question #32 : Multiplying And Dividing Radicals

Simplify.

$\frac{3}{1-\sqrt{2}}$

Possible Answers:

$-3-3\sqrt{2}$

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

$3-3\sqrt{2}$

$3+3\sqrt{2}$

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

Correct answer:

$-3-3\sqrt{2}$

Explanation:

$\frac{3}{1-\sqrt{2}}$ To get rid of the radical in the denominator, we need to multiply top and bottom by the conjugate which is the opposite sign of the radical expression. This would be $1+\sqrt{2}$ .

$\frac{3}{1-\sqrt{2}}\cdot\frac{1+\sqrt{2}}{1+\sqrt{2}}=\frac{3+3\sqrt{2}}{1-2}=\frac{3+3\sqrt{2}}{-1}$ Remember to use FOIL when multiplying out the denominators. Now, with out answer, we can distribute out the .

$\frac{3+3\sqrt{2}}{-1}=-3-3\sqrt{2}$

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Example Question #33 : Multiplying And Dividing Radicals

Multiply: $6\sqrt{3}*2\sqrt{8}$

Possible Answers:

$24\sqrt{6}$

Cannot combine these radicals

$12\sqrt{6}$

$12\sqrt{24}$

None of these

Correct answer:

$24\sqrt{6}$

Explanation:

$6\sqrt{3}*2\sqrt{8}$

Multiply the outer numbers first:

$12(\sqrt{3}*\sqrt{8})$

Combine radicals:

$12\sqrt{24}$

Simplify the radical:

$12\sqrt{6*4}$

$\mathbf{24\sqrt{6}}$

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Example Question #133 : Simplifying Radicals

$\sqrt{3}*\sqrt{7}$

Possible Answers:

$\sqrt{37}$

$\sqrt{21}$

$\sqrt{10}$

Correct answer:

$\sqrt{21}$

Explanation:

When multiplying radicals, we can just multiply the values inside the radicand.

$\sqrt{3}*\sqrt{7}=\sqrt{3*7}=\sqrt{21}$

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Algebra II : Simplifying Radicals

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #25 : Multiplying And Dividing Radicals

Example Question #25 : Multiplying And Dividing Radicals

Example Question #29 : Multiplying And Dividing Radicals

Example Question #30 : Multiplying And Dividing Radicals

Example Question #131 : Simplifying Radicals

Example Question #132 : Simplifying Radicals

Example Question #33 : Multiplying And Dividing Radicals

Example Question #32 : Multiplying And Dividing Radicals

Example Question #33 : Multiplying And Dividing Radicals

Example Question #133 : Simplifying Radicals

All Algebra II Resources

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