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High School Math : Exponents

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Example Questions

Example Question #1 : Understanding Negative Exponents

Which of the following is equivalent to $3^{-2}$ ?

Possible Answers:

$\frac{1}{6}$

$\frac{1}{9}$

Correct answer:

$\frac{1}{9}$

Explanation:

By definition,

$b^{-x} = \frac{1}{b^{x}}$ .

In our problem, b = 3 and x = 2 .

Then, we have $\frac{1}{3^{2}} = \frac{1}{9}$ .

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Example Question #3 : Negative Exponents

Solve for :

$(x+5)^{-3} = -1$

Possible Answers:

Correct answer:

Explanation:

Raise both sides of the equation to the inverse power of to cancel the exponent on the left hand side of the equation.

$\rightarrow ((x+5)^{-3})^{-\frac{1}{3}} = (-1)^{-\frac{1}{3}}$

$\rightarrow x+5 = -1$

Subtract from both sides:

$\rightarrow (x+5) - 5 = (-1)-5$

$\rightarrow x = -6$

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Example Question #2 : Fractional Exponents

Convert the exponent to radical notation.

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

Possible Answers:

$\small \small \sqrt[3]{x^7}$

$\small \frac{1}{x^4}$

$\small \small \sqrt[7]{x^3}$

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

Correct answer:

$\small \small \sqrt[7]{x^3}$

Explanation:

Remember that exponents in the denominator refer to the root of the term, while exponents in the numerator can be treated normally.

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

$x^{\frac{3}{7}}=\sqrt[7]{x^3}$

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Example Question #1 : Understanding Exponents

Which of the following is equivalent to $64^{\frac{1}{2}}$ ?

Possible Answers:

$\frac{1}{32}$

Correct answer:

Explanation:

By definition, a number raised to the $\frac{1}{2}$ power is the same as the square root of that number.

Since the square root of 64 is 8, 8 is our solution.

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Example Question #1 : Fractional Exponents

Simplify the expression:

$\small (16^{\frac{1}{2}})(256^{\frac{3}{4}})$

Possible Answers:

1024

256

Correct answer:

256

Explanation:

Remember that fraction exponents are the same as radicals.

$\small 16^{\frac{1}{2}}=\sqrt{16}=4$

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

A shortcut would be to express the terms as exponents and look for opportunities to cancel.

$16^{\frac{1}{2}}=(4^2)^{\frac{1}{2}}=4$

$\small 256^{\frac{3}{4}}=(4^4)^{\frac{3}{4}}=4^3=64$

Either method, we then need to multiply to two terms.

$\small (4)(64)=256$

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Example Question #1 : Simplifying Exponents

Simplify the following expression.

$\frac{4x^3y^8z^2}{8x^6y^2z^4}$

Possible Answers:

$\frac{y^4}{2x^2z^2}$

$\frac{y^6}{2x^3z^2}$

$\frac{4y^6}{8x^3z^2}$

$\frac{4y^4}{8x^2z^2}$

Correct answer:

$\frac{y^6}{2x^3z^2}$

Explanation:

When dividing with exponents, the exponent in the denominator is subtracted from the exponent in the numerator. For example: $\frac{x^3}{x^6}= x^{3-6}=x^{-3}=\frac{1}{x^3}$ .

In our problem, each term can be treated in this manner. Remember that a negative exponent can be moved to the denominator.

$\frac{4x^3y^8z^2}{8x^6y^2z^4}=\frac{4}{8}x^{-3}y^6z^{-2}=\frac{4y^6}{8x^3z^2}$

Now, simplifly the numerals.

$\frac{4y^6}{8x^3z^2}=\frac{y^6}{2x^3z^2}$

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Example Question #1 : Simplifying Exponents

Simplify the following expression.

$2^{3} \cdot 2^{6}$

Possible Answers:

$4^{9}$

$2^{9}$

$4^{18}$

$2^{18}$

$6^{6}$

Correct answer:

$2^{9}$

Explanation:

We are given: $2^{3} \cdot 2^{6}$ .

Recall that when we are multiplying exponents with the same base, we keep the base the same and add the exponents.

Thus, we have $2^{3 + 6} = 2^{9}$ .

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Example Question #1 : Simplifying Exponents

Simplify the following expression.

$\frac{2^{9}}{2^{11}}$

Possible Answers:

$\frac{18}{22}$

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

$2^{\frac{9}{11}}$

$\frac{9}{11}$

$2^{2}$

Correct answer:

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

Explanation:

Recall that when we are dividing exponents with the same base, we keep the base the same and subtract the exponents.

Thus, we have $\frac{2^{9}}{2^{11}} = 2^{9 - 11} = 2^{-2}$ .

We also recall that for negative exponents,

$b^{-x} = \frac{1}{b^{x}}$ .

Thus, $2^{-2} = \frac{1}{2^{2}}$ .

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Example Question #1 : Multiplying And Dividing Exponents

Simplify the following exponent expression:

$(\frac{a^{\frac{-3}{2}}}{3^6b^{\frac{-2}{3}}})^{\frac{-1}{2}}$

Possible Answers:

$\frac{a^{\frac{3}{4}}}{b^{\frac{1}{3}}}$

$\frac{27a^{\frac{3}{4}}}{b^{\frac{1}{3}}}$

$\frac{9a^{\frac{3}{4}}}{b^{\frac{1}{3}}}$

$\frac{3a^{\frac{3}{4}}}{b^{\frac{1}{3}}}$

$\frac{18a^{\frac{3}{4}}}{b^{\frac{1}{3}}}$

Correct answer:

$\frac{27a^{\frac{3}{4}}}{b^{\frac{1}{3}}}$

Explanation:

Begin by rearranging the terms in the numerator and denominator so that the exponents are positive:

$(\frac{a^{\frac{-3}{2}}}{3^6b^{\frac{-2}{3}}})^{\frac{-1}{2}}$

$(\frac{3^6b^{\frac{-2}{3}}}{a^{\frac{-3}{2}}})^{\frac{1}{2}}$

$(\frac{3^6a^{\frac{3}{2}}}{b^{\frac{2}{3}}})^{\frac{1}{2}}$

Multiply the exponents:

$(\frac{3^3a^{\frac{3}{4}}}{b^{\frac{1}{3}}})$

Simplify:

$\frac{27a^{\frac{3}{4}}}{b^{\frac{1}{3}}}$

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Example Question #1 : Simplifying Exponents

Simplify the expression:

$x^3 + \frac{2x^2}{x^{-1}}$

Possible Answers:

$x^3+\frac{2}{x^3}$

x^3+2x^2

3x^3

x^3+2x

x^3+2

Correct answer:

3x^3

Explanation:

First simplify the second term, and then combine the two:

$x^3 + \frac{2x^2}{x^{-1}}$

$= x^3 + 2x^{2-(-1)} = x^3 + 2x^3 = 3x^3$

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High School Math : Exponents

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Understanding Negative Exponents

Example Question #3 : Negative Exponents

Example Question #2 : Fractional Exponents

Example Question #1 : Understanding Exponents

Example Question #1 : Fractional Exponents

Example Question #1 : Simplifying Exponents

Example Question #1 : Simplifying Exponents

Example Question #1 : Simplifying Exponents

Example Question #1 : Multiplying And Dividing Exponents

Example Question #1 : Simplifying Exponents

All High School Math Resources

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