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Algebra II : Algebra II

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10 Diagnostic Tests 630 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #49 : Understanding Exponents

Evaluate $21^{-2}$

Possible Answers:

$\frac{1}{441}$

441

$-\frac{1}{441}$

Correct answer:

$\frac{1}{441}$

Explanation:

When expressing negative exponents, we rewrite as such:

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

in which is the positive exponent raising base .

$21^{-2}=\frac{1}{21^2}=\frac{1}{441}$

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Example Question #50 : Understanding Exponents

Evaluate $-4^{-4}$

Possible Answers:

$\frac{1}{16}$

$-\frac{1}{256}$

$\frac{1}{256}$

$-\frac{1}{16}$

Correct answer:

$-\frac{1}{256}$

Explanation:

When expressing negative exponents, we rewrite as such:

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

in which is the positive exponent raising base .

$-4^{-4}=-\frac{1}{4^4}=-\frac{1}{256}$

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Example Question #51 : Negative Exponents

Evaluate $(-5)^{-4}$

Possible Answers:

$-\frac{1}{625}$

$\frac{1}{20}$

$\frac{1}{625}$

Correct answer:

$\frac{1}{625}$

Explanation:

When expressing negative exponents, we rewrite as such:

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

in which is the positive exponent raising base .

$(-5)^{-4}=\frac{1}{(-5)^4}=\frac{1}{625}$

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Example Question #51 : Negative Exponents

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

Possible Answers:

$\frac{1}{8}$

$-\frac{1}{6}$

$-\frac{1}{8}$

Correct answer:

Explanation:

When expressing negative exponents, we rewrite as such:

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

in which is the positive exponent raising base .

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

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Example Question #53 : Understanding Exponents

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

Possible Answers:

$\frac{\sqrt{6}}{6}$

$\sqrt{6}$

$-\sqrt{6}$

$\frac{1}{36}$

Correct answer:

$\sqrt{6}$

Explanation:

When dealing with fractional exponents, we rewrite as such:

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

in which is the index of the radical and is the exponent raising base .

When expressing negative exponents, we rewrite as such:

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

in which is the positive exponent raising base .

$\frac{1}{6}^{-\frac{1}{2}}=\frac{1}{(\sqrt\frac{1}{6})}=\frac{1}{\frac{1}{\sqrt{6}}}=\sqrt{6}$

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Example Question #51 : Exponents

Evaluate $12^{-\frac{1}{2}}$

Possible Answers:

$\frac{\sqrt{3}}{6}$

$\frac{1}{12}$

$2\sqrt{3}$

$6\sqrt{3}$

$\frac{\sqrt{3}}{2}$

Correct answer:

$\frac{\sqrt{3}}{6}$

Explanation:

When dealing with fractional exponents, we rewrite as such:

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

in which is the index of the radical and is the exponent raising base .

When expressing negative exponents, we rewrite as such:

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

in which is the positive exponent raising base .

$12^{-\frac{1}{2}}=\frac{1}{(\sqrt{12})}=\frac{1}{2{\sqrt{3}}}=\frac{\sqrt{3}}{6}$

Remember when getting rid of radicals, just multiply top and bottom by that radical.

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Example Question #55 : Understanding Exponents

Evaluate $48^{-\frac{2}{3}}$

Possible Answers:

$\frac{\sqrt[3]{6}}{24}$

$\frac{\sqrt{3}}{24}$

$\frac{\sqrt{3}}{6}$

$\frac{\sqrt[3]{36}}{16}$

$\frac{\sqrt{6}}{16}$

Correct answer:

$\frac{\sqrt[3]{6}}{24}$

Explanation:

When dealing with fractional exponents, we rewrite as such:

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

in which is the index of the radical and is the exponent raising base .

When expressing negative exponents, we rewrite as such:

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

in which is the positive exponent raising base .

$48^{-\frac{2}{3}}=\frac{1}{\sqrt[3]{48^2}}=\frac{1}{\sqrt[3]{2304}}$

Let's find a perfect cube which is

64 =(4^3) .

$\frac{1}{\sqrt[3]{2304}}=\frac{1}{\sqrt[3]{64}*\sqrt[3]{36}}=\frac{1}{4\sqrt[3]{36}}$

To simplify, we need to multiply top and bottom by an appropriate cubic root. We know 36=6^2 so if we multiply top and bottom by $\sqrt[3]{6}$ we will get an integer in the denominator.

$\frac{1}{4\sqrt[3]{36}}*\frac{\sqrt[3]{6}}{\sqrt[3]{6}}=\frac{\sqrt[3]{6}}{4*6}=\frac{\sqrt[3]{6}}{24}$

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Example Question #56 : Understanding Exponents

Evaluate $12^{-\frac{3}{4}}$

Possible Answers:

$\frac{\sqrt[4]{12}}{8}$

$\frac{\sqrt[4]{12}}{12}$

$\frac{\sqrt[3]{36}}{16}$

$\frac{\sqrt{6}}{12}$

$\frac{\sqrt[3]{12}}{12}$

Correct answer:

$\frac{\sqrt[4]{12}}{12}$

Explanation:

When dealing with fractional exponents, we rewrite as such:

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

in which is the index of the radical and is the exponent raising base .

When expressing negative exponents, we rewrite as such:

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

in which is the positive exponent raising base .

$12^{-\frac{3}{4}}=\frac{1}{\sqrt[4]{12^3}}=\frac{1}{\sqrt[4]{1728}}$

Let's find a perfect fourth power which is

16 = (2^4) .

$\frac{1}{\sqrt[4]{1728}}=\frac{1}{\sqrt[4]{16}*\sqrt[4]{108}}=\frac{1}{2\sqrt[4]{108}}$

To simplify, we need to multiply top and bottom by an appropriate fourth root. We know 108=3^3*2^2 .

We need to complete the numbers to the fourth power. It we multiply top and bottom by $\sqrt[4]{3*2^2}=\sqrt[4]{12}$ we will get an integer in the denominator.

$\frac{1}{2\sqrt[4]{108}}*\frac{\sqrt[4]{12}}{\sqrt[4]{12}}=\frac{\sqrt[4]{12}}{2*6}=\frac{\sqrt[4]{12}}{12}$

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Example Question #57 : Understanding Exponents

Simplify $(4x^{2}y^{-6})(2x^{-7}y^{^3})$ .

Possible Answers:

$\frac{1}{8x^{5}y^{3}}$

$\frac{1}{8x^{14}y^{18}}$

$\frac{8}{x^{14}y^{18}}$

$8x^{5}y^{3}$

$\frac{8}{x^{5}y^{3}}$

Correct answer:

$\frac{8}{x^{5}y^{3}}$

Explanation:

First multiply the like terms, remembering that when multiplying terms that have exponents, you add the exponents.

$(4x^{2}y^{-6})(2x^{-7}y^{3})=(4\cdot 2)(x^{2}\cdot x^{-7})(y^{-6}\cdot y^{3})$

$=8x^{2-7}y^{-6+3}$

$=8x^{-5}y^{-3}$

Negative exponents indicate that the term should be in the denominator, so the final answer is:

$\frac{8}{x^{5}y^{3}}$

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Example Question #58 : Understanding Exponents

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

Possible Answers:

$\frac{5}{36}$

$\sqrt{6}+\sqrt{3}$

$\frac{1}{81}$

$\frac{5}{18}$

Correct answer:

$\frac{5}{36}$

Explanation:

Convert each negative exponent into fractional form.

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

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

Simplify the denominators.

$\frac{1}{36}+\frac{1}{9}$

Convert the second fraction with a common denominator of 36.

$\frac{1}{36}+\frac{4}{36}= \frac{5}{36}$

The answer is: $\frac{5}{36}$

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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #49 : Understanding Exponents

Example Question #50 : Understanding Exponents

Example Question #51 : Negative Exponents

Example Question #51 : Negative Exponents

Example Question #53 : Understanding Exponents

Example Question #51 : Exponents

Example Question #55 : Understanding Exponents

Example Question #56 : Understanding Exponents

Example Question #57 : Understanding Exponents

Example Question #58 : Understanding Exponents

All Algebra II Resources

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