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Algebra II : Understanding Exponents

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Example Questions

Example Question #641 : Mathematical Relationships And Basic Graphs

Write $x^{2/5}$ in radical form.

Possible Answers:

$5\sqrt{x}$

$2\sqrt[5]{x}$

$\frac{1}{\sqrt[5]{x^{2}}}$

$\sqrt[5]{x^{2}}$

$\sqrt[]{x^{5}}$

Correct answer:

$\sqrt[5]{x^{2}}$

Explanation:

Fractional exponents are an alternate way to write a radical expression. The numerator of the fractional exponent becomes the exponent of the term inside the radical and the denominator of the fractional exponent determines the index of the radical.

So $x^{2/5}$ becomes $\sqrt[5]{x^{2}}$ .

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Example Question #171 : Understanding Exponents

Simplify $7x^{\frac{2}{3}}\cdot x^{\frac{1}{4}}$ .

Possible Answers:

$7x^{\frac{3}{7}}$

$7x^{\frac{11}{12}}$

$7x^{\frac{1}{6}}$

This cannot be simplified.

$8x^{\frac{11}{12}}$

Correct answer:

$7x^{\frac{11}{12}}$

Explanation:

When multiplying terms with exponents, you must add the exponents.

In order to add the exponents, you need to find a common denominator, in this case the common denominator is 12.

$7x^{\frac{2}{3}}\cdot x^{\frac{1}{4}}=7x^{\frac{8}{12}}\cdot x^{\frac{3}{12}}=7x^{\frac{11}{12}}$

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Example Question #55 : Fractional Exponents

Simplify $(216x^{6})^{-\frac{4}{3}}$ .

Possible Answers:

$\frac{1}{(x\sqrt[4]{216x^{2}})^{3}}$

$1296x^{8}$

$\frac{1}{1296x^{8}}$

$\frac{1}{1296x^{6}}$

$1296x^{6}$

Correct answer:

$\frac{1}{1296x^{8}}$

Explanation:

First, eliminate the negative exponent by placing the term in the denominator:

$(216x^{6})^{-\frac{4}{3}}=\frac{1}{(216x^{6})^{\frac{4}{3}}}$

With fractional exponents, the denominator is the index of the radical and the numerator is the power the term is raised to:

$\frac{1}{(216x^{6})^{\frac{4}{3}}}=\frac{1}{(\sqrt[3]{216x^{6}})^{4}}$

Take the cube root of the denominator and raise to the 4th power:

$\frac{1}{(6x^{2})^{4}}=\frac{1}{1296x^{8}}$

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Example Question #56 : Fractional Exponents

Simplify $(16x^{8})^{\frac{5}{2}}$ .

Possible Answers:

$1024x^{9}$

$4x^{20}$

$64x^{20}$

$1024x^{20}$

$\sqrt[5]{4096x^{16}}$

Correct answer:

$1024x^{20}$

Explanation:

With a fractional exponent, the denominator tells you the index of the radical and the numerator tells you what power to raise the term to. So this problem can be rewritten as:

$(16x^{8})^{\frac{5}{2}}=(\sqrt{16x^{8}})^{5}$

Now take the square root and raise to the 5th power:

$(\sqrt{16x^{8}})^{5}=(4x^{^{4}})^{5}=1024x^{20}$

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Example Question #57 : Fractional Exponents

Solve: $2^{-3}\times 3^{-2}$

Possible Answers:

$\frac{1}{36}$

$\frac{1}{7776}$

$\frac{1}{32768}$

$\frac{1}{144}$

$\frac{1}{72}$

Correct answer:

$\frac{1}{72}$

Explanation:

Before we can multiply the terms, convert both terms with negative exponents to fractions.

$x^{-n}=\frac{1}{x^n}$

Rewrite the terms and simplify.

$2^{-3}\times 3^{-2} = \frac{1}{2^3}\times\frac{1}{3^2} = \frac{1}{8}\times \frac{1}{9}$

Multiply the denominators together.

The answer is: $\frac{1}{72}$

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Example Question #172 : Exponents

Simplify: $4^{\frac{3}{2}}+ 8^{\frac{2}{3}}$

Possible Answers:

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

Correct answer:

Explanation:

Evaluate the first term by rewriting the term out as a product of exponents.

$4^{\frac{3}{2}} =(4^3)^{\frac{1}{2}} = 64^{\frac{1}{2}}$

The power of a half is similar to the square root of the number.

$64^{\frac{1}{2}} = \sqrt{64} = 8$

Evaluate the second term.

$8^{\frac{2}{3}} = (8^{\frac{1}{3}})^2 = (\sqrt[3]{8})^2 = 2^2 =4$

Add both numbers.

8+4=12

The answer is:

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Example Question #59 : Fractional Exponents

Evaluate $8^{\frac{2}{5}}$

Possible Answers:

$\sqrt[5]{64}$

$10\sqrt{2}$

$\sqrt[5]{8}$

$\sqrt{8^5}$

$2\sqrt[5]{8}$

Correct answer:

$\sqrt[5]{64}$

Explanation:

When dealing with fractional exponents, we write as

$x^\frac{a}{b}=\sqrt[b]{x^a}$

which is the index of the radical, is the exponent raising base .

Therefore we have

$8^\frac{2}{5}=\sqrt[5]{8^2}=\sqrt[5]{64}$

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Example Question #53 : Fractional Exponents

Evaluate $6^{-\frac{1}{2}}$

Possible Answers:

$\sqrt6$

$-\sqrt6$

$-\frac{\sqrt6}{6}$

$\frac{\sqrt6}{6}$

$-\frac{1}{6}$

Correct answer:

$\frac{\sqrt6}{6}$

Explanation:

When dealing with fractional exponents, we write as

$x^\frac{a}{b}=\sqrt[b]{x^a}$

which is the index of the radical, is the exponent raising base .

We evaluate negative exponents as

$x^{-a}=\frac{1}{x^a}$

which is the positive exponent raising base .

Therefore,

$6^{-\frac{1}{2}}=\frac{1}{\sqrt6}=\frac{1*\sqrt6}{\sqrt6*\sqrt6}=\frac{\sqrt6}{6}$

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Example Question #3311 : Algebra Ii

Evaluate $\frac{1}{4}^{-\frac{3}{4}}$

Possible Answers:

$2\sqrt[3]{4}$

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

$2\sqrt[4]{4}$

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

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

Correct answer:

$2\sqrt[4]{4}$

Explanation:

When dealing with fractional exponents, we write as

$x^\frac{a}{b}=\sqrt[b]{x^a}$

which is the index of the radical, is the exponent raising base .

We evaluate negative exponents as

$x^{-a}=\frac{1}{x^a}$

which is the positive exponent raising base .

Therefore

$\frac{1}{4}^{-\frac{3}{4}}=\frac{1}{\frac{1}{\sqrt[4]{4^3}}}=\sqrt[4]{64}$ .

We can factor out 2^4 or .

$\sqrt[4]{64}=\sqrt[4]{16}*\sqrt[4]{4}=2*\sqrt[4]{4}=2\sqrt[4]{4}$

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Example Question #61 : Fractional Exponents

Simplify, if possible: $(\frac{3}{4})^{\frac{4}{3}}$

Possible Answers:

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

$\frac{3\sqrt[3]{6}}{8}$

$\frac{3\sqrt[3]{6}}{4}$

$\frac{2\sqrt[3]{6}}{3}$

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

Correct answer:

$\frac{3\sqrt[3]{6}}{8}$

Explanation:

In order to simplify this, we will need to multiply both the numerator and denominator with the outside power according to the power rule of exponents.

$(\frac{3}{4})^{\frac{4}{3}} = \frac{3^\frac{4}{3}}{4^\frac{4}{3}}$

The numerator of the power represents the power raised to. The denominator of the power represents the root of the radical. Rewrite the fractions.

$\frac{3^\frac{4}{3}}{4^\frac{4}{3}} = \frac{(\sqrt[3]{3})^4}{(\sqrt[3]{4})^4} = \frac{\sqrt[3]{3}\cdot \sqrt[3]{3}\cdot \sqrt[3]{3}\cdot \sqrt[3]{3}}{\sqrt[3]{4}\cdot \sqrt[3]{4}\cdot \sqrt[3]{4}\cdot \sqrt[3]{4}}$

Recall that multiplying the radicals in the third root three times will leave the integer by itself.

The expression becomes:

$\frac{\sqrt[3]{3}\cdot \sqrt[3]{3}\cdot \sqrt[3]{3}\cdot \sqrt[3]{3}}{\sqrt[3]{4}\cdot \sqrt[3]{4}\cdot \sqrt[3]{4}\cdot \sqrt[3]{4}}= \frac{3 \sqrt[3]{3}}{4 \sqrt[3]{4}}$

Rationalize the denominator by multiplying the top and bottom by $\sqrt[3]{4}$ twice in order to eliminate the cube root radical denominator.

$\frac{3 \sqrt[3]{3}}{4 \sqrt[3]{4}} \cdot \frac{\sqrt[3]{4}}{\sqrt[3]{4}} \cdot \frac{\sqrt[3]{4}}{\sqrt[3]{4}}= \frac{3 \cdot \sqrt[3]{48}}{4(4)} = \frac{3 \sqrt[3]{48}}{16}$

Simplify $\sqrt[3]{48}$ .

$\sqrt[3]{48} = \sqrt[3]{8\cdot 6} = \sqrt[3]{8} \cdot \sqrt[3]{6} = 2 \sqrt[3]{6}$

Replace the new term and reduce the fraction. The fraction becomes:

$\frac{3 \sqrt[3]{48}}{16} = \frac{3 \cdot 2 \sqrt[3]{6}}{16} = \frac{3\sqrt[3]{6}}{8}$

The answer is: $\frac{3\sqrt[3]{6}}{8}$

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Algebra II : Understanding Exponents

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #641 : Mathematical Relationships And Basic Graphs

Example Question #171 : Understanding Exponents

Example Question #55 : Fractional Exponents

Example Question #56 : Fractional Exponents

Example Question #57 : Fractional Exponents

Example Question #172 : Exponents

Example Question #59 : Fractional Exponents

Example Question #53 : Fractional Exponents

Example Question #3311 : Algebra Ii

Example Question #61 : Fractional Exponents

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