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Algebra II : Understanding Exponents

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #1 : Simplifying Expressions

Simplify:

$[x^{1/2}]^{7/3}$

Possible Answers:

$x^{8/6}$

$x^{5/2}$

$x^{7/6}$

$x^{17/6}$

$x^{4/3}$

Correct answer:

$x^{7/6}$

Explanation:

$[x^{m}]^{n}=x^{m\times n}$

$x^{1/2\times 7/3}=x^{7/6}$

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Example Question #2 : Fractional Exponents

Write the product of $\small a^\frac{3}{4}*a^\frac{3}{8}*a^\frac{5}{2}$ in radical form

Possible Answers:

$\small \sqrt[2]{a^5}$

$\small \sqrt[8]{a^2^9}$

$\small \sqrt[a]{8^2^9}$

$\small \sqrt[a]{29^8}$

$\small \sqrt[29]{a^8}$

Correct answer:

$\small \sqrt[8]{a^2^9}$

Explanation:

This problem relies on the key knowledge that $\small x^\frac{a}{b}=\sqrt[b]{x^a}$ and that the multiplying terms with exponents requires adding the exponents. Therefore, we can rewrite the expression thusly:

$\small a^\frac{3}{4}*a^\frac{3}{8}*a^\frac{5}{2}=a^{\frac{3}{4}+\frac{3}{8}+\frac{5}{2}}=a^{\frac{29}{8}}=\sqrt[8]{a^{29}}$

Therefore, $\small \sqrt[8]{a^{29}}$ is our final answer.

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Example Question #1 : Fractional Exponents

Evaluate the following expression:

$\bigg(\frac{27}{125}\bigg)^{\frac{1}{3}}$

Possible Answers:

$\frac{3}{\sqrt{25}}$

$\frac{3}{5}$

$\frac{9}{5}$

$\sqrt{\frac{27}{125}}$

$\frac{3}{25}$

Correct answer:

$\frac{3}{5}$

Explanation:

$\bigg(\frac{27}{125}\bigg)^{\frac{1}{3}} = \frac{27^{\frac{1}{3}}}{125^{\frac{1}{3}}}=\frac{3}{5}$

$\bigg(\frac{27}{125}\bigg)^{\frac{1}{3}}= \sqrt[3]{\frac{27}{125}}=\frac{\sqrt[3]{27}}{\sqrt[3]{125}}=\frac{3}{5}$

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Example Question #2 : Fractional Exponents

Simplify:

$\frac{x^{\frac{1}{2}}y^2}{x^{\frac{-3}{2}}y^4}$

Possible Answers:

$\frac{x^2}{y^2}$

$\frac{1}{x^2y^2}$

$\frac{y^2}{x^2}$

x^2y^2

Correct answer:

$\frac{x^2}{y^2}$

Explanation:

Keep in mind that when you are dividing exponents with the same base, you will want to subtract the exponent found in the denominator from the exponent found in the numerator.

To find the exponent for , subtract the denominator's exponent from the numerator's exponent.

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

To find the exponent for , subtract the denominator's exponent from the numerator's exponent.

2-4=-2

Since the exponent is negative, you will want to put the in the denominator in order to make it positive.

So then,

$\frac{x^{\frac{1}{2}}y^2}{x^{\frac{-3}{2}}y^4}=\frac{x^2}{y^2}$

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Example Question #122 : Exponents

Find the value of $9^{\frac{3}{2}}$ .

Possible Answers:

729

Correct answer:

Explanation:

When you have a number or value with a fractional exponent,

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

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

So then,

$9^{\frac{3}{2}}=\sqrt[2]{9^3}=\sqrt{729}=27$

$9^{\frac{3}{2}}=(\sqrt9)^3=3^3=27$

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Example Question #123 : Exponents

Find the value of $64^{\frac{4}{3}}$

Possible Answers:

512

4096

1024

256

Correct answer:

256

Explanation:

When you have a number or value with a fractional exponent,

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

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

So then,

$64^{\frac{4}{3}}=\sqrt[3]{64^4}=\sqrt[3]{2^{24}}=2^8=256$

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Example Question #3 : Fractional Exponents

Simplify:

$(27t^3)^\frac{5}{3}$

Possible Answers:

243t^5

$243t^{\frac{8}{3}}$

81t^5

27t^5

Correct answer:

243t^5

Explanation:

When exponents are raised to another exponent, you will need to multiply the exponents together.

When you have a number or value with a fractional exponent,

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

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

So,

$27^{\frac{5}{3}}=\sqrt[3]{27^5}=\sqrt[3]{3^{15}}=3^5=243$

$(t^3)^{\frac{5}{3}}=t^{3\times\frac{5}{3}}=t^{\frac{15}{3}}=t^5$

$(27t^3)^\frac{5}{3}=243t^5$

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Example Question #121 : Exponents

Simplify: $25^{\frac{3}{2}}$

Possible Answers:

125

Correct answer:

125

Explanation:

An option to solve this is to split up the fraction. Rewrite the fractional exponent as follows:

$25^{\frac{3}{2}} = (25^{\frac{1}{2}})^3$

A value to its half power is the square root of that value.

$25^{\frac{1}{2}} = \sqrt{25}=5$

Substitute this value back into $(25^{\frac{1}{2}})^3$ .

$(25^{\frac{1}{2}})^3 = (5)^3 = 5\times 5\times 5 = 125$

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Example Question #2 : Rational Exponents

Which of the following is equivalent to $b^{\frac{4}{5}}$ ?

Possible Answers:

5b^4

$\sqrt[4]{b^5}$

$\sqrt[5]{b^4}$

$\sqrt[4]{b^{-5}}$

Correct answer:

$\sqrt[5]{b^4}$

Explanation:

Which of the following is equivalent to $b^{\frac{4}{5}}$ ?

When dealing with fractional exponents, keep the following in mind: The numerator is making the base bigger, so treat it like a regular exponent. The denominator is making the base smaller, so it must be the root you are taking.

This means that $b^{\frac{4}{5}}$ is equal to the fifth root of b to the fourth. Perhaps a bit confusing, but it means that we will keep b^4 , but put the whole thing under $\sqrt[5]{b}$ .

So if we put it together we get:

$\sqrt[5]{b^4}$

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Example Question #11 : Fractional Exponents

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

Possible Answers:

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

$\sqrt{2}$

$\frac{1}{4}$

Correct answer:

$\sqrt{2}$

Explanation:

When dealing with fractional exponents, remember this form:

$\sqrt[a]{x^n}$ is the index of the radical which is also the denominator of the fraction, represents the base of the exponent, and is the power the base is raised to. That value is the numerator of the fraction.

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

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Algebra II : Understanding Exponents

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #1 : Simplifying Expressions

Example Question #2 : Fractional Exponents

Example Question #1 : Fractional Exponents

Example Question #2 : Fractional Exponents

Example Question #122 : Exponents

Example Question #123 : Exponents

Example Question #3 : Fractional Exponents

Example Question #121 : Exponents

Example Question #2 : Rational Exponents

Example Question #11 : Fractional Exponents

All Algebra II Resources

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