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Algebra II : Exponents

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Example Question #61 : Understanding Exponents

Evaluate: $(\frac{4}{5})^{-2}$

Possible Answers:

$\frac{25}{8}$

$\frac{8}{25}$

$\frac{2\sqrt5}{5}$

$\frac{16}{25}$

$\frac{25}{16}$

Correct answer:

$\frac{25}{16}$

Explanation:

Rewrite the negative exponent as a fraction.

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

Following this rule, rewrite the given expression.

$(\frac{4}{5})^{-2}=\frac{1}{(\frac{4}{5})^2} = \frac{1}{\frac{4}{5} \cdot \frac{4}{5}}= \frac{1}{\frac{16}{25}}$

Take the reciprocal of the denominator to eliminate the complex fraction.

The answer is: $\frac{25}{16}$

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Example Question #62 : Understanding Exponents

Simplify: $2^{-1}+3^{-2}$

Possible Answers:

$\frac{2}{27}$

$\frac{5}{18}$

$\frac{11}{18}$

$\sqrt2 +\sqrt3$

$\frac{1}{36}$

Correct answer:

$\frac{11}{18}$

Explanation:

To eliminate the negative exponent, we will need to convert the terms to a fraction.

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

According to this rule, convert both terms.

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

Simplify the fractions.

$\frac{1}{2}+\frac{1}{9} = \frac{9}{18}+ \frac{2}{18} =\frac{11}{18}$

The answer is: $\frac{11}{18}$

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Example Question #63 : Understanding Exponents

Simplify: $\frac{6^{-3}}{8^{-3}}$

Possible Answers:

$\frac{4}{3}$

$\frac{27}{64}$

$\frac{64}{27}$

$\frac{3}{4}$

$\frac{4096}{729}$

Correct answer:

$\frac{64}{27}$

Explanation:

Convert the numerator and denominator to fractions by using the formula below.

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

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

Rewrite the fractions using a division sign.

$\frac{\frac{1}{6^3}}{\frac{1}{8^3}} =\frac{1}{6^3} \div\frac{1}{8^3}$

Take the reciprocal of the second term and change the division sign to a multiplication symbol.

$\frac{1}{6^3} \times 8^3 = \frac{8^3}{6^3} = \frac{8\cdot 8\cdot 8}{6\cdot 6\cdot 6} =\frac{4\cdot 4\cdot 4}{3\cdot 3\cdot 3} = \frac{64}{27}$

The answer is: $\frac{64}{27}$

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Example Question #64 : Understanding Exponents

Evaluate the exponents: $\frac{2^{-1}+3^{-1}}{2^{-2}-3^{-2}}$

Possible Answers:

Correct answer:

Explanation:

Write the rule for negative exponents.

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

Convert the powers on the numerator and denominator.

$\frac{2^{-1}+3^{-1}}{2^{-2}-3^{-2}} = \frac{\frac{1}{2}+\frac{1}{3}}{\frac{1}{4}-\frac{1}{9}}$

Add the numerator. Find the least common denominator and add the numerators.

$\frac{1}{2}+\frac{1}{3} = \frac{3}{6}+\frac{2}{6} = \frac{5}{6}$

Subtract the denominator also by converting to the least common denominator.

$\frac{1}{4}-\frac{1}{9} = \frac{9}{36}-\frac{4}{36} = \frac{5}{36}$

Divide the numerator with the denominator. Convert the complex fraction by rewriting it using a division sign. Swap the numerator and denominator of the second term and change the division sign to a multiplication sign.

$\frac{\frac{5}{6}}{\frac{5}{36}} = \frac{5}{6}\div \frac{5}{36} = \frac{5}{6}\times \frac{36}{5} = 6$

The answer is:

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Example Question #65 : Understanding Exponents

Simplify: $(\frac{3}{4^{-2}})^{-2}$

Possible Answers:

$\frac{256}{3}$

$\frac{256}{9}$

$\frac{3}{256}$

2304

$\frac{1}{2304}$

Correct answer:

$\frac{1}{2304}$

Explanation:

Whenever we have a negative exponent, we can convert that to a fraction.

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

Evaluate the inner term.

$(\frac{3}{4^{-2}})^{-2} = (\frac{3}{\frac{1}{4^2}})^{-2} = (\frac{3}{\frac{1}{16}})^{-2}$

The complex fraction can be reduced as follows:

$\frac{3}{\frac{1}{16}} = 3\div \frac{1}{16} = 3\times \frac{16}{1} = 48$

Substitute this value into the parentheses.

$(48)^{-2} = \frac{1}{48^2} = \frac{1}{2304}$

The answer is: $\frac{1}{2304}$

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Example Question #66 : Understanding Exponents

Simplify: $(\frac{2}{7})^{-2}$

Possible Answers:

$\frac{49}{2}$

$\frac{2}{49}$

$\frac{49}{4}$

$\frac{4}{49}$

$\frac{7}{4}$

Correct answer:

$\frac{49}{4}$

Explanation:

Convert the negative exponent as follows:

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

Rewrite the expression without the negative exponent.

$(\frac{2}{7})^{-2}=\frac{1}{(\frac{2}{7})^{2}}=\frac{1}{(\frac{4}{49})}$

Rewrite the complex fraction using a division sign.

$1\div\frac{4}{49} = 1\times\frac{49}{4} = \frac{49}{4}$

The answer is: $\frac{49}{4}$

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Example Question #67 : Understanding Exponents

Simplify $7^{-6}$

Possible Answers:

$\frac{6}{7}$

$\frac{1}{7^6}$

$-\frac{1}{7^6}$

$\frac{1}{42}$

$-\frac{1}{42}$

Correct answer:

$\frac{1}{7^6}$

Explanation:

We evaluate negative exponents as

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

which is the positive exponent raising base .

Therefore

$7^{-6}=\frac{1}{7^6}$ .

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Example Question #61 : Exponents

Simplify $\frac{1}{3}^{-3}$

Possible Answers:

$\frac{1}{27}$

$-\frac{1}{27}$

Correct answer:

Explanation:

We evaluate negative exponents as

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

which is the positive exponent raising base .

Therefore

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

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Example Question #69 : Understanding Exponents

Determine the value of: $4^{-2}-2^{-6}$

Possible Answers:

$\frac{1}{32}$

$\frac{1}{16}$

$\frac{3}{32}$

$\frac{3}{64}$

$\frac{1}{12}$

Correct answer:

$\frac{3}{64}$

Explanation:

The negative exponents can be rewritten as a fraction.

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

Convert the terms.

$4^{-2}-2^{-6} = \frac{1}{4^2}-\frac{1}{2^6} = \frac{1}{16}- \frac{1}{64}$

Convert the first fraction so that the denominator are similar.

$\frac{1}{16}- \frac{1}{64} = \frac{1(4)}{16(4)}- \frac{1}{64} = \frac{4}{64}-\frac{1}{64}$

Subtract the numerators.

The answer is: $\frac{3}{64}$

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Example Question #70 : Understanding Exponents

Simplify: $(\frac{3}{5})^{-2}$

Possible Answers:

$\frac{9}{25}$

$\frac{25}{9}$

$\frac{1}{3}$

$\frac{3\sqrt5}{5}$

$\frac{3}{25}$

Correct answer:

$\frac{25}{9}$

Explanation:

To eliminate the negative exponent, use the following property:

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

Rewrite the expression using this property.

$(\frac{3}{5})^{-2} = \frac{1}{(\frac{3}{5})^2} = \frac{1}{(\frac{3}{5})(\frac{3}{5})} = \frac{1}{\frac{9}{25}}$

Simplify the complex fraction.

$\frac{1}{\frac{9}{25}} =1\div \frac{9}{25} =1\times \frac{25} {9} =\frac{25}{9}$

The answer is: $\frac{25}{9}$

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Algebra II : Exponents

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #61 : Understanding Exponents

Example Question #62 : Understanding Exponents

Example Question #63 : Understanding Exponents

Example Question #64 : Understanding Exponents

Example Question #65 : Understanding Exponents

Example Question #66 : Understanding Exponents

Example Question #67 : Understanding Exponents

Example Question #61 : Exponents

Example Question #69 : Understanding Exponents

Example Question #70 : Understanding Exponents

All Algebra II Resources

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