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Algebra II : Understanding Exponents

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #3292 : Algebra Ii

Simplify: $9^{\frac{1}{9}}$

Possible Answers:

$\sqrt[9]{9}$

$-\sqrt[9]{9}$

$\frac{1}{\sqrt[9]{9}}$

Correct answer:

$\sqrt[9]{9}$

Explanation:

When dealing with fractional exponents, we rewrite as such: $x^{\frac{a}{b}}=\sqrt[b]{x^a}$ which is the index of the radical and is the exponent raising base .

$9^{\frac{1}{9}}=\sqrt[9]{9}$

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Example Question #3293 : Algebra Ii

Simplify: $16^{-\frac{2}{3}}$

Possible Answers:

$\sqrt[3]{256}$

$\frac{1}{\sqrt[3]{256}}$

$-\frac{1}{\sqrt[3]{256}}$

$-\sqrt[3]{256}$

Correct answer:

$\frac{1}{\sqrt[3]{256}}$

Explanation:

When dealing with fractional exponents, we rewrite as such: $x^{\frac{a}{b}}=\sqrt[b]{x^a}$ which is the index of the radical and is the exponent raising base . When dealing with negative exponents, we convert to fractions as such: $x^{-a}=\frac{1}{x^a}$ which is the positive exponent raising base .

$16^{-\frac{2}{3}}=\frac{1}{\sqrt[3]{16^2}}=\frac{1}{\sqrt[3]{256}}$

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Example Question #3294 : Algebra Ii

Simplify: $20^{-\frac{2}{5}}$

Possible Answers:

$\sqrt[5]{400}$

$-\frac{1}{8}$

$\frac{1}{\sqrt[5]{400}}$

$-\sqrt[5]{400}$

Correct answer:

$\frac{1}{\sqrt[5]{400}}$

Explanation:

$20^{-\frac{2}{5}}=\frac{1}{\sqrt[5]{20^2}}=\frac{1}{\sqrt[5]{400}}$

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Example Question #161 : Understanding Exponents

Simplify: $\frac{1}{4}^{\frac{1}{2}}$

Possible Answers:

$\frac{1}{2}$

$\frac{1}{12}$

$\frac{1}{8}$

$\frac{1}{16}$

Correct answer:

$\frac{1}{2}$

Explanation:

When dealing with fractional exponents, we rewrite as such: $x^{\frac{a}{b}}=\sqrt[b]{x^a}$ which is the index of the radical and is the exponent raising base .

$\frac{1}{4}^{\frac{1}{2}}=\sqrt{\frac{1}{4}}=\frac{1}{2}$

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Example Question #3296 : Algebra Ii

Simplify:

$a^{\frac{1}{4}}$

where a=3x^4y^8

Possible Answers:

$xy\sqrt[4]{3}$

$xy^2\sqrt[4]{3}$

$x^2y^4\sqrt[4]{3}$

$81x^{16}y^{32}$

Correct answer:

$xy^2\sqrt[4]{3}$

Explanation:

When dealing with fractional exponents, remember that the numerator of the fraction represents the power to which we are taking the term that has the exponent, and the denominator represents the degree of the root we are taking of that term.

For our expression, the numerator is 1, which means we raise a to the first power. The denominator is 4, which means we are taking the fourth root of the term:

$\sqrt[4]{(3x^4y^8)^1}$

We can only move the cubes out of the radical, and when we do so, we get

$xy^2\sqrt[4]{3}$

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Example Question #3297 : Algebra Ii

Evaluate $17^\frac{1}{2}$

Possible Answers:

$\sqrt{17}$

$\sqrt{8.5}$

8.5

Correct answer:

$\sqrt{17}$

Explanation:

When dealing with fractional exponents, we rewrite as such:

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

in which is the index of the radical and is the exponent raising base .

$17^\frac{1}{2}=\sqrt{17}$

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Example Question #3298 : Algebra Ii

Evaluate $18^\frac{2}{3}$

Possible Answers:

$\sqrt[3]{54}$

$\sqrt[3]{324}$

$8\sqrt{2}$

$\sqrt{5832}$

Correct answer:

$\sqrt[3]{324}$

Explanation:

When dealing with fractional exponents, we rewrite as such:

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

in which is the index of the radical and is the exponent raising base .

$18^\frac{2}{3}=\sqrt[3]{18^2}=\sqrt[3]{324}$

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Example Question #3299 : Algebra Ii

Evaluate $24^\frac{2}{5}$

Possible Answers:

9.6

$\sqrt[5]{676}$

$2\sqrt[5]{18}$

$2\sqrt[5]{2}$

$24\sqrt{2}$

Correct answer:

$2\sqrt[5]{18}$

Explanation:

When dealing with fractional exponents, we rewrite as such:

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

in which is the index of the radical and is the exponent raising base .

$24^\frac{2}{5}=\sqrt[5]{24^2}=\sqrt[5]{576}=2\sqrt[5]{18}$

We were able to simplify it by factoring out perfect fifth root.

In this case, it was 32 =2^5 .

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Example Question #51 : Fractional Exponents

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

Possible Answers:

$\frac{1}{4}$

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

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

$\sqrt{2}$

$-\sqrt{2}$

Correct answer:

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

Explanation:

When dealing with fractional exponents, we rewrite as such:

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

in which is the index of the radical and is the exponent raising base .

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

Remember when getting rid of radicals, we just multiply top and bottom by the radical.

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Example Question #52 : Fractional Exponents

Simplify:

$(9x)^{\frac{2}{3}}+(10x)^{\frac{3}{2}}$

Possible Answers:

$3\sqrt[3]{3x^2}+10x\sqrt{10x}$

$3\sqrt[3]{3x^2}-10x\sqrt{10x}$

$3x\sqrt[3]{3x}+10x\sqrt{10x}$

$3\sqrt[3]{9x^2}+10x\sqrt{10x}$

Correct answer:

$3\sqrt[3]{3x^2}+10x\sqrt{10x}$

Explanation:

To simplify fractional exponents, remember that the numerator of the fraction corresponds to the power the term is taken, and the denominator indicates what root we are taking of that term (i.e. 2 means square root, 3 means cube root and so on).

Doing this, we get

$\sqrt[3]{(9x)^2}+\sqrt{(10x)^3}$

Expanding the inside of the roots, we get

$\sqrt[3]{81x^2}+\sqrt{1000x^3}$

We can now factor the terms on the inside, pulling out any cubes or squares, respectively:

$\sqrt[3]{27\cdot 3\cdot x^2} +\sqrt{100\cdot 10\cdot x^2\cdot x}=3\sqrt[3]{3x^2}+10x\sqrt{10x}$

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Algebra II : Understanding Exponents

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #3292 : Algebra Ii

Example Question #3293 : Algebra Ii

Example Question #3294 : Algebra Ii

Example Question #161 : Understanding Exponents

Example Question #3296 : Algebra Ii

Example Question #3297 : Algebra Ii

Example Question #3298 : Algebra Ii

Example Question #3299 : Algebra Ii

Example Question #51 : Fractional Exponents

Example Question #52 : Fractional Exponents

All Algebra II Resources

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