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Algebra II : Radicals as Exponents

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #82 : Radicals

Simplify: $32^{\frac{2}{5}} -16^{-\frac{1}{4}}$

Possible Answers:

$\frac{3}{8}$

$\frac{9}2$

$\frac{8}{3}$

$\frac{4}{5}$

$\frac{7}{2}$

Correct answer:

$\frac{7}{2}$

Explanation:

Simplify each term. Rewrite the fractional powers as radicals. The denominator is the indicator of the root, and the numerator is the power.

$32^{\frac{2}{5}} =( \sqrt[5]{32})^2 = (2)^2 = 4$

$16^{-\frac{1}{4}} = \frac{1}{16^{\frac{1}{4}}} = \frac{1}{\sqrt[4]{16}} = \frac{1}{2}$

Subtract the simplified numbers.

$4-\frac{1}{2} = \frac{8}{2}-\frac{1}{2}= \frac{7}{2}$

The answer is: $\frac{7}{2}$

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Example Question #83 : Radicals

Simplify: $100^{\frac{3}{2}}$

Possible Answers:

1000

$10\sqrt{30}$

$\sqrt[3]{100}$

$\sqrt[3]{10}$

3000

Correct answer:

1000

Explanation:

The denominator of the exponent represents the root. The numerator represents the power of the term.

$100^{\frac{3}{2}} =( \sqrt{100})^3$

Simplify by order of operations.

$(10)^3 = 10\cdot 10\cdot 10=1000$

The answer is: 1000

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Example Question #81 : Understanding Radicals

Evaluate: $-(32)^{\frac{3}{5}}$

Possible Answers:

$\textup{No solution.}$

Correct answer:

Explanation:

The numerator of the fractional power indicates the power the quantity is raised to.

The denominator represents the root.

Rewrite as a radical and simplify.

The fifth root asks for a number that multiplies by itself five times to get the number inside the radical.

$-(32)^{\frac{3}{5}} = -(\sqrt[5]{32})^3 = -(2)^3 = -8$

The answer is:

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Example Question #81 : Radicals

Evaluate: $(\frac{125}{27})^\frac{2}{3}$

Possible Answers:

$\frac{9}{5}$

$\frac{25}{3}$

$\frac{25}{9}$

$\frac{16}{9}$

Correct answer:

$\frac{25}{9}$

Explanation:

The power can be distributed to both terms in the fraction.

$(\frac{125}{27})^\frac{2}{3} = (\frac{125^\frac{2}{3}}{27^\frac{2}{3}})$

The denominator of the power represents the root, while the numerator represent that power that the quantity is raised to.

Rewrite the terms.

$125^\frac{2}{3}=(\sqrt[3]{125})^2 = 5^2 = 25$

$27^\frac{2}{3} = (\sqrt[3]{27})^2 = (3)^2=9$

Divide the two terms.

The answer is: $\frac{25}{9}$

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Example Question #31 : Radicals As Exponents

What is the value of: $2\sqrt{\sqrt{32}}$ ?

Possible Answers:

$4 \sqrt[4]{3}$

$8\sqrt[4]{2}$

$4 \sqrt[4]{2}$

$2\sqrt[4]{4}$

$2 \sqrt[4]{2}$

Correct answer:

$4 \sqrt[4]{2}$

Explanation:

In order to solve this expression, we will need to rewrite each square root using one half as the exponent. This will allow us to solve for the value of the inner terms.

$2\sqrt{\sqrt{32}}=2((\sqrt{32})^\frac{1}{2})^\frac{1}{2} = 2\sqrt[4]{32}$

We can then rewrite the radical using factors of perfect numbers raised to the fourth power, such as: 1^4 =1, 2^4 =16

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

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

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Example Question #81 : Understanding Radicals

Evaluate: $(\frac{1}{8})^{\frac{1}{3}}$

Possible Answers:

$\frac{2\sqrt2}{3}$

$\sqrt2$

$\frac{\sqrt2}{2}$

$\frac{1}{2}$

Correct answer:

$\frac{1}{2}$

Explanation:

The exponent can be distributed through both terms in the parentheses.

$(\frac{1}{8})^{\frac{1}{3}} = \frac{1^{\frac{1}{3}}}{8^{\frac{1}{3}} } = \frac{\sqrt[3]{1}}{\sqrt[3]{8}}$

Simplify each radical. The cube root is identifying a number that multiplies itself three times to produce the value inside the radical.

The answer is: $\frac{1}{2}$

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Example Question #82 : Understanding Radicals

Which of the following is equivalent to $2x^{\frac{3}{8}}$ ?

Possible Answers:

$2\sqrt[3]{x^8}$

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

$2\sqrt[8]{x^3}$

$\sqrt[8]{2x^3}$

$\sqrt[3]{2x^8}$

Correct answer:

$2\sqrt[8]{x^3}$

Explanation:

The denominator of the powered term represents the root of the radical. The numerator of the fraction is the power that the quantity of the radical is raised by.

Note that the integer in front of the x-term will not be included inside the radical.

The answer is: $2\sqrt[8]{x^3}$

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Example Question #89 : Radicals

Evaluate: $(\frac{2}{3})^{\frac{1}{2}}$

Possible Answers:

$\frac{\sqrt2}{3}$

$\frac{\sqrt6}{3}$

$2\sqrt6$

$\frac{\sqrt2}{6}$

$\sqrt3$

Correct answer:

$\frac{\sqrt6}{3}$

Explanation:

The exponent can be distributed through the numerator and denominator.

$(\frac{2}{3})^{\frac{1}{2}}=\frac{2^{\frac{1}{2}}}{3^{\frac{1}{2}}}$

Rewrite both the top and bottom with radicals. The power of one-half is the same as taking the square root of the number.

$\frac{\sqrt2}{\sqrt3}$

Rationalize the denominator by multiplying both the top and bottom by the denominator.

$\frac{\sqrt2}{\sqrt3}\cdot \frac{\sqrt3}{\sqrt3} =\frac{\sqrt6}{3}$

The answer is: $\frac{\sqrt6}{3}$

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Example Question #90 : Radicals

Evaluate: $-16^\frac{5}{2}$

Possible Answers:

1024

-1024i

1024i

$\textup{There is no solution.}$

Correct answer:

Explanation:

The denominator of the exponential term represents the root of a radical. The numerator of the fraction represents the power that the quantity is raised to.

$-16^\frac{5}{2} = -(\sqrt{16})^5$

Be careful not to include the negative term since the fractional exponent only counts the sixteen, and not the entire quantity.

$-(\sqrt{16})^5 = -(4)^5 = -(4)(4)(4)(4)(4) = -1024$

The answer is:

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Example Question #91 : Understanding Radicals

Solve: $-16^{-\frac{3}{4}}$

Possible Answers:

$-\frac{1}{8}$

$\frac{1}{8}$

$-\frac{1}{4}$

$\frac{1}{4}$

$-\frac{1}{16}$

Correct answer:

$-\frac{1}{8}$

Explanation:

We will need to rewrite the negative exponent as a fraction.

$-16^{-\frac{3}{4}} = -\frac{1}{16^{\frac{3}{4}}}$

Recall that the denominator of the power represents the root index, and the numerator of the power represents what the quantity is raised to.

$16^{\frac{3}{4}} =( \sqrt[4]{16})^3 = (2)^3 = 8$

Replace the denominator.

$-\frac{1}{16^{\frac{3}{4}}} = -\frac{1}{8}$

The answer is: $-\frac{1}{8}$

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Algebra II : Radicals as Exponents

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #82 : Radicals

Example Question #83 : Radicals

Example Question #81 : Understanding Radicals

Example Question #81 : Radicals

Example Question #31 : Radicals As Exponents

Example Question #81 : Understanding Radicals

Example Question #82 : Understanding Radicals

Example Question #89 : Radicals

Example Question #90 : Radicals

Example Question #91 : Understanding Radicals

All Algebra II Resources

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