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Algebra II : Understanding Exponents

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #71 : Understanding Exponents

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

Possible Answers:

$\frac{25}{4}$

$\frac{5}{24}$

$\frac{5}{12}$

$\frac{5}{2}$

$\frac{25}{2}$

Correct answer:

$\frac{25}{4}$

Explanation:

Simplify the first term. Rewrite the negative exponent as a fraction.

$\frac{1}{6^{-1}} = \frac{1}{\frac{1}{6}}$

Rewrite the complex fraction.

$\frac{1}{\frac{1}{6}} = 1\div \frac{6}{1} = 6$

Simplify the second term.

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

Add the terms.

$\frac{1}{6^{-1}}+2^{-2} =6+ \frac{1}{4} = 6\frac{1}{4}= \frac{25}{4}$

The answer is: $\frac{25}{4}$

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Example Question #71 : Negative Exponents

Evaluate: $-3^{-3}-2^{-2}-1^{-1}$

Possible Answers:

$\frac{131}{108}$

$-\frac{61}{27}$

$-\frac{51}{4}$

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

$-\frac{139}{108}$

Correct answer:

$-\frac{139}{108}$

Explanation:

For negative exponents, convert all terms into fractional form by the following property.

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

$-3^{-3}-2^{-2}-1^{-1} = -\frac{1}{3^3}-\frac{1}{2^2}-\frac{1}{1^1}$

The fractions become:

$-\frac{1}{27}-\frac{1}{4}-1$

Identify the least common denominator of these numbers and convert all numbers with the same LCD.

$27\times 4 =108$

The LCD is 108.

$-\frac{1}{27}-\frac{1}{4}-1 = -\frac{4}{108}-\frac{27}{108}-\frac{108}{108}$

The answer is: $-\frac{139}{108}$

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Example Question #71 : Negative Exponents

Solve: $\frac{(-2+4)^{-4}}{6}$

Possible Answers:

$\frac{8}{3}$

$\frac{1}{48}$

$\frac{1}{96}$

$\frac{1}{16}$

$\frac{3}{8}$

Correct answer:

$\frac{1}{96}$

Explanation:

Eliminate the negative exponent by the following property:

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

Simplify and rewrite the expression.

$\frac{(-2+4)^{-4}}{6}=\frac{(2)^{-4}}{6}=\frac{\frac{1}{2^{4}}}{6}= \frac{\frac{1}{16}}{6}$

Simplify the complex fraction.

$\frac{\frac{1}{16}}{6} = \frac{1}{16} \div 6= \frac{1}{16} \times \frac{1}{6} = \frac{1}{96}$

The answer is: $\frac{1}{96}$

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Example Question #71 : Negative Exponents

Evaluate: $\frac{3}{3^{-3}}+\frac{2}{2^{-2}}$

Possible Answers:

$\frac{11}{18}$

$\frac{89}{648}$

$\frac{5}{31}$

Correct answer:

Explanation:

Simplify the negative exponents by converting the terms into fractions.

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

Evaluate each term of $\frac{3}{3^{-3}}+\frac{2}{2^{-2}}$ .

$\frac{3}{3^{-3}} =\frac{3}{(\frac{1}{3})^{3}}= \frac{3}{\frac{1}{27}}$

Simplify the complex fraction.

$3\div \frac{1}{27} = 3\times 27 = 81$

Evaluate the second term.

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

$2 \div \frac{1}{4} = 2 \times 4 = 8$

$\frac{3}{3^{-3}}+\frac{2}{2^{-2}} = 81+8 =89$

The answer is:

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Example Question #75 : Understanding Exponents

Solve: $6^{-2}\times 2^{-3}$

Possible Answers:

$\frac{1}{72}$

$\frac{1}{60}$

$\frac{1}{144}$

$\frac{1}{288}$

$\frac{1}{17}$

Correct answer:

$\frac{1}{288}$

Explanation:

Rewrite the negative exponents as fractions using the following property.

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

$6^{-2}\times 2^{-3}=\frac{1}{6^2}\times \frac{1}{2^3} = \frac{1}{36}\times \frac{1}{8}$

Multiply the denominators together.

The answer is: $\frac{1}{288}$

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Example Question #76 : Understanding Exponents

Simplify: $(ab^2c)^{-3}$

Possible Answers:

$\frac{1}{a^3bc^3}$

$\frac{1}{a^3b^6c^3}$

$\frac{b^2}{a^3c^3}$

$\frac{b}{a^3c^3}$

$\frac{b^6}{a^3c^3}$

Correct answer:

$\frac{1}{a^3b^6c^3}$

Explanation:

The terms as a quantity can be rewritten by the following property:

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

Rewrite the expression.

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

Expand the terms by distributing the third power with every term in the denominator.

The answer is: $\frac{1}{a^3b^6c^3}$

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Example Question #71 : Exponents

Simplify: 3x^-^2

Possible Answers:

$\frac{1}{9x^2}$

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

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

9x^2

Correct answer:

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

Explanation:

When we find negative exponents in an expression, we move only what is raised to the negative power to the bottom of the fraction.

In this case, only the x was raised to the (-2) power, so it is moved to the bottom of the fraction.

When this has been moved to the bottom of the fraction, it becomes a positive 2.

*Remember, there should be no negative exponents in a final answer.

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Example Question #78 : Understanding Exponents

Solve: $10^{-2}+10^{-3}$

Possible Answers:

$\frac{10}{1101}$

$\frac{10}{101}$

$\frac{10}{111}$

$\frac{11}{1000}$

$\frac{11}{100}$

Correct answer:

$\frac{11}{1000}$

Explanation:

Convert both terms in negative exponents to fractional form.

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

$10^{-2}+10^{-3}=\frac{1}{100}+\frac{1}{1000}$

Add the two fractions by converting the first fraction to a common denominator.

$\frac{1}{100}+\frac{1}{1000} = \frac{10}{1000}+\frac{1}{1000}= \frac{11}{1000}$

The answer is: $\frac{11}{1000}$

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Example Question #3211 : Algebra Ii

Evaluate: $9^{-1}+3^{-3}$

Possible Answers:

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

$\frac{244}{27}$

$\frac{82}{27}$

$\frac{4}{27}$

$\frac{2}{9}$

Correct answer:

$\frac{4}{27}$

Explanation:

The negative exponents can be simplified as follows:

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

Rewrite the expression.

$9^{-1}+3^{-3} = \frac{1}{9}+\frac{1}{3^3}= \frac{1}{9}+\frac{1}{27}$

Convert the one-ninth to $\frac{3}{27}$ to match denominators, and add the fractions.

$\frac{3}{27}+\frac{1}{27}= \frac{4}{27}$

The answer is: $\frac{4}{27}$

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Example Question #3212 : Algebra Ii

Simplify: $-6-3^{-2}$

Possible Answers:

$-\frac{1}{18}$

$-\frac{55}{9}$

$-\frac{35}{6}$

$-\frac{37}{6}$

$-\frac{53}{9}$

Correct answer:

$-\frac{55}{9}$

Explanation:

The negative exponent can be rewritten into a fraction.

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

Convert the expression.

$-6-3^{-2}= -6-\frac{1}{3^2} = -6- \frac{1}{9} = -6\frac{1}{9}$

Ignoring the negative sign, multiply the denominator with the whole number and add the numerator will give an improper fraction. The denominator stays the same.

$-\frac{(9\times 6)+1}{9}$

The answer is: $-\frac{55}{9}$

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Algebra II : Understanding Exponents

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #71 : Understanding Exponents

Example Question #71 : Negative Exponents

Example Question #71 : Negative Exponents

Example Question #71 : Negative Exponents

Example Question #75 : Understanding Exponents

Example Question #76 : Understanding Exponents

Example Question #71 : Exponents

Example Question #78 : Understanding Exponents

Example Question #3211 : Algebra Ii

Example Question #3212 : Algebra Ii

All Algebra II Resources

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