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College Algebra : Review and Other Topics

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Example Questions

Example Question #300 : College Algebra

Solve: $2(5^{-3})$

Possible Answers:

$\frac{2}{125}$

$-\frac{2}{125}$

$-\frac{1}{30}$

$\frac{1}{1000}$

Correct answer:

$\frac{2}{125}$

Explanation:

Evaluate the negative exponential first.

$5^{-3} =\frac{1}{5^3} = \frac{1}{125}$

Replace the fraction.

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

The answer is: $\frac{2}{125}$

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Example Question #21 : Review And Other Topics

What is the value of $(\frac{2}{3})^4$ ?

Possible Answers:

$\frac{16}{81}$

$\frac{4}{3}$

$\frac{8}{81}$

$\frac{4}{15}$

Correct answer:

$\frac{16}{81}$

Explanation:

A value raised to the fourth power will multiply itself four times.

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

The answer is: $\frac{16}{81}$

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Example Question #22 : Integer Exponents

Evaluate: 2^6+3^4-5^3

Possible Answers:

Correct answer:

Explanation:

Step 1: Evaluate each term separately...

2^6=2*2*2*2*2*2=64

3^4=3*3*3*3=81

5^3=5*5*5=125

Step 2: Re-write statement using the converted numbers.

64+81-125

Add:

145-125

Simplify:

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Example Question #22 : Review And Other Topics

Express $5^{-3}$ with only positive exponents.

Possible Answers:

$\frac {1}{5^3}$

No Solution

$\frac {1}{5^2}$

$\frac {1}{625}$

Correct answer:

$\frac {1}{5^3}$

Explanation:

Whenever we have a negative exponent in a exponential term, we change the term..

We write divided by the term to make the exponent positive..

So.. $5^{-3}=\frac {1}{5^3}$

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Example Question #24 : Integer Exponents

Evaluate: 2^2-3^2+16

Possible Answers:

Correct answer:

Explanation:

Step 1: Expand each term:

2^2=2*2, 3^2=3*3, 16=16

Step 2: Multiply to find the value of each term..

2^2=4,3^2=9, 16

Step 3: Re-write the original question, replacing the exponents with the numbers we calculated...

4-9+16

Step 4: Add/Subtract

4-9=-5

-5+16=11

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Example Question #23 : Review And Other Topics

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 #24 : Review And Other Topics

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

Possible Answers:

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

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

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

5b^4

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 #25 : Review And Other Topics

Evaluate $\frac{1}{4}^{-\frac{1}{2}}$

Possible Answers:

$\frac{1}{16}$

$\frac{1}{4}$

$-\frac{1}{16}$

Correct answer:

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.

With a negative exponent, we need to remember this form:

$\frac{1}{x^n}$ represents the base of the exponent, and is the power in a positive value.

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

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Example Question #26 : Review And Other Topics

$\left ( \frac{125}{343} \right )^{\frac{2}{3}}$

Possible Answers:

$\frac{250}{686}$

$\frac{5}{343}$

$\frac{5}{7}$

$\frac{25}{49}$

$\frac{625}{2401}$

Correct answer:

$\frac{25}{49}$

Explanation:

First, distribute the exponent to both the numerator and denominator of the fraction.

$\frac{125^{\frac{2}{3}}}{343^{\frac{2}{3}}}$

The numerator of a fractional exponent is the power you take the number to and the denominator is the root that you take the number to.

$\frac{\sqrt[3]{125^{2}}}{\sqrt[3]{343^{2}}}$

You can take the cubed root and square the numbers in either order but if you can do the root first that is often easier.

$\frac{5^{2}}{7^{2}}$

$\frac{25}{49}$

This is the answer. Alternatively, you could have squared the numbers first before taking the cubed root.

$\frac{\sqrt[3]{15625}}{\sqrt[3]{117649}}$

$\frac{25}{49}$

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Example Question #27 : Review And Other Topics

Evaluate:

$\left(\frac{1}{256}\right)^{-\frac{1}{4}}$

Possible Answers:

$-\frac{1}{4}$

Correct answer:

Explanation:

In order to evaluate fractional exponents, we can express them using the following relationship:

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

In this formula, represents the index of the radical from the denominator of the fraction and is the exponent that raises the base: . When exponents are negative, we can express them using the following relationship:

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

We can then rewrite and solve the expression in the following way.

$\left(\frac{1}{256}\right)^{-\frac{1}{4}}=\frac{1}{\frac{1}{\sqrt[4]{256}}}=\frac{1}{\frac{1}{4}}$

Dividing by a fraction is the same as multiplying by its reciprocal.

$\frac{1}{\frac{1}{4}}=1\times \frac{4}{1}=4$

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College Algebra : Review and Other Topics

Study concepts, example questions & explanations for College Algebra

All College Algebra Resources

Example Questions

Example Question #300 : College Algebra

Example Question #21 : Review And Other Topics

Example Question #22 : Integer Exponents

Example Question #22 : Review And Other Topics

Example Question #24 : Integer Exponents

Example Question #23 : Review And Other Topics

Example Question #24 : Review And Other Topics

Example Question #25 : Review And Other Topics

Example Question #26 : Review And Other Topics

Example Question #27 : Review And Other Topics

All College Algebra Resources

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