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Algebra II : Mathematical Relationships and Basic Graphs

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Example Questions

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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Example Question #59 : Understanding Exponents

Evaluate: $2^{-3x}=8$

Possible Answers:

$-\frac{1}{4}$

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

$-\frac{8}{3}$

$-\frac{2}{3}$

Correct answer:

Explanation:

In order to determine the value of x, we will need to convert the base of the right side similar to the left.

Eight is similar to two cubed. Rewrite the equation.

$2^{-3x}=2^3$

Now that our bases are the same, we can set the exponents equal to each other.

-3x=3

Divide by negative three on both sides.

$\frac{-3x}{-3}=\frac{3}{-3}$

The answer is: x=-1

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Example Question #60 : Understanding Exponents

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

Possible Answers:

$\frac{1}{27x}$

27x

$\frac{x}{3}$

$\frac{1}{9x}$

Correct answer:

27x

Explanation:

The negative exponent can be converted into a fraction.

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

Rewrite the fraction.

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

Rewrite the complex fraction using a division sign.

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

Turn the division sign to a multiplication sign and take the reciprocal of the second term.

$3x\times \frac{3^2}{1}=3x(9) = 27x$

The answer is: 27x

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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 #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

Example Question #59 : Understanding Exponents

Example Question #60 : Understanding Exponents

All Algebra II Resources

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