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Algebra II : Mathematical Relationships and Basic Graphs

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Example Questions

Example Question #111 : Negative Exponents

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

Possible Answers:

$\frac{43}{24}$

$\frac{19}{18}$

$\frac{1}{6}$

$\frac{31}{24}$

$\frac{13}{72}$

Correct answer:

$\frac{31}{24}$

Explanation:

Simplify each term. The negative exponents can be rewritten as a fraction.

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

$\frac{2^{-3}}{3^{-2}} = \frac{\frac{1}{2^3}}{\frac{1}{3^2}}= \frac{\frac{1}{8}}{\frac{1}{9}}= \frac{1}{8}\div \frac{1}{9} = \frac{1}{8}\times \frac{9}{1} =\frac{9}{8}$

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

Add the fractions.

$\frac{9}{8}+\frac{1}{6} = \frac{9(3)}{8(3)}+\frac{1(4)}{6(4)} = \frac{31}{24}$

The answer is: $\frac{31}{24}$

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Example Question #112 : Negative Exponents

Solve: $\frac{1}{12}+6^{-1}-6^{-2}$

Possible Answers:

$\frac{7}{26}$

$\frac{5}{36}$

$\frac{2}{9}$

$\frac{8}{9}$

$\frac{4}{9}$

Correct answer:

$\frac{2}{9}$

Explanation:

Convert each negative exponent to a fraction. Use the rule below to rewrite the terms.

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

$\frac{1}{12}+6^{-1}-6^{-2}=\frac{1}{12}+\frac{1}{6^{1}}-\frac{1}{6^{2}}$

The expression becomes:

$\frac{1}{12}+\frac{1}{6}-\frac{1}{36}$

Convert the denominators to a common LCD.

$\frac{1(3)}{12(3)}+\frac{1(6)}{6(6)}-\frac{1}{36} = \frac{3}{36} +\frac{6}{36} -\frac{1}{36}=\frac{8}{36}$

Reduce this fraction.

The answer is: $\frac{2}{9}$

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Example Question #113 : Negative Exponents

Simplify the exponents: $\frac{3^{-3}}{\frac{1}{9}}$

Possible Answers:

$\frac{1}{9}$

243

$\frac{1}{27}$

$\frac{1}{3}$

Correct answer:

$\frac{1}{3}$

Explanation:

The negative exponent can be rewritten as a fraction.

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

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

Divide this fraction with one-ninth.

$\frac{\frac{1}{27}}{\frac{1}{9}}= \frac{1}{27}\div \frac{1}{9} = \frac{1}{27}\times \frac{9}{1}$

Reduce the fraction.

The answer is: $\frac{1}{3}$

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Example Question #114 : Negative Exponents

Solve: $\frac{7^{-1}}{5^{-2}}$

Possible Answers:

$\frac{7}{25}$

$\frac{25}{7}$

$\frac{25}{49}$

175

$\frac{1}{175}$

Correct answer:

$\frac{25}{7}$

Explanation:

The negative exponents must be rewritten in fractional form by the following rule.

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

This expression becomes:

$\frac{7^{-1}}{5^{-2}} = \frac{\frac{1}{7^{1}}}{\frac{1}{5^{2}}} = \frac{\frac{1}{7}}{\frac{1}{25}}$

Rewrite this complex fraction using a division sign. Take the reciprocal of the second term, and change the division sign to multiplication.

$\frac{1}{7}\div \frac{1}{25}=\frac{1}{7}\times \frac{25}{1}$

The answer is: $\frac{25}{7}$

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Example Question #115 : Negative Exponents

Solve: $\frac{(\frac{2}{3})^{-3}}{2^{-1}}$

Possible Answers:

$\frac{4}{9}$

$\frac{16}{27}$

$\frac{27}{4}$

$\frac{4}{3}$

Correct answer:

$\frac{27}{4}$

Explanation:

Evaluate both terms on the numerator and denominator separately.

We can rewrite the negative exponents as the reciprocal of the positive exponent.

Use the multiplicative rule of division to simplify the complex fraction.

$(\frac{2}{3})^{-3} = \frac{1}{(\frac{2}{3})^3} = 1\div (\frac{8}{27}) = 1\times \frac{27}{8}= \frac{27}{8}$

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

Divide the two fractions.

$\frac{27}{8}\div \frac{1}{2} = \frac{27}{8}\times \frac{2}{1} =\frac{27}{4}$

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

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Example Question #116 : Negative Exponents

Simplify: $[(2^{-3})\cdot (5^{-2})]^{-1}$

Possible Answers:

200

$\frac{1}{200}$

$\frac{1}{100}$

$\frac{8}{25}$

$\frac{25}{8}$

Correct answer:

200

Explanation:

The negative exponents can be rewritten as fractions based on the following rule:

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

Rewrite all the terms in the parentheses.

$[(2^{-3})\cdot (5^{-2})]^{-1} =[(\frac{1}{2^3})\cdot (\frac{1}{5^{2}})]^{-1} =[(\frac{1}{8})\cdot (\frac{1}{25})]^{-1}$

Simplify the expression.

$[\frac{1}{200}]^{-1} = 1\div \frac{1}{200} = 1\times \frac{200}{1} = 200$

The answer is: 200

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Example Question #117 : Negative Exponents

Simplify: $\frac{3^{-2}}{8}$

Possible Answers:

$\frac{1}{36}$

$\frac{1}{72}$

$\frac{1}{17}$

$\frac{1}{48}$

$\frac{1}{24}$

Correct answer:

$\frac{1}{72}$

Explanation:

The negative exponent can be rewritten as follows:

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

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

Rewrite the expression.

$\frac{3^{-2}}{8} =\frac{ \frac{1}{9}}{8} = \frac{1}{9}\times \frac{1}{8} = \frac{1}{72}$

The answer is: $\frac{1}{72}$

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Example Question #3242 : Algebra Ii

Simplify the following expression:

$\frac{4x^{-2}y^{3}}{18x^{-4}y^{2}}$

Possible Answers:

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

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

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

$\frac{x^{6}y^{5}}{4}$

Correct answer:

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

Explanation:

$\frac{4x^{-2}y^3}{18x^{-4}y^{2}}=\frac{4x^{4}y^{3}}{18x^{2}y^{2}}=\left (\frac{4}{18} \right )x^{4-2}y^{3-2}=\left (\frac{4}{18} \right )x^{2}y=\frac{2x^{2}y}{9}$

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Example Question #1 : Fractional Exponents

Simplify the expression:

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

Possible Answers:

256

1024

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 : Understanding Fractional Exponents

Convert the exponent to radical notation.

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

Possible Answers:

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

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

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

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

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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Algebra II : Mathematical Relationships and Basic Graphs

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #111 : Negative Exponents

Example Question #112 : Negative Exponents

Example Question #113 : Negative Exponents

Example Question #114 : Negative Exponents

Example Question #115 : Negative Exponents

Example Question #116 : Negative Exponents

Example Question #117 : Negative Exponents

Example Question #3242 : Algebra Ii

Example Question #1 : Fractional Exponents

Example Question #1 : Understanding Fractional Exponents

All Algebra II Resources

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