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Algebra II : Mathematical Relationships and Basic Graphs

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #21 : Distributing Exponents (Power Rule)

$\left (\frac{3}{4} \right )^{4}$

Possible Answers:

$\frac{81}{256}$

$\frac{9}{12}$

$\frac{3}{4}$

Correct answer:

$\frac{81}{256}$

Explanation:

When you have an exponent outside of parentheses, it means it is necessary to distribute it to all parts of whatever numbers or variables are inside the parentheses.

Distribute the exponent 4 to both the 3 and the 4 inside the parentheses

$\frac{3^{4}}{4^{4}}$

Now take 3 to the fourth power (or 3 four times) in the numerator and 4 to the fourth power (or 4 four times) in the denominator

$\frac{3\cdot 3\cdot 3\cdot 3}{4\cdot 4\cdot 4\cdot 4}$

$\frac{81}{256}$

This fraction does not reduce so this is the final answer.

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Example Question #24 : Distributing Exponents (Power Rule)

Simplify $\left ( 4x^{3}y^{2} \right )^{4}$

Possible Answers:

$4x^{7}y^{6}$

$256x^{12}y^{8}$

$256x^{7}y^{6}$

$16x^{7}y^{6}$

$4x^{12}y^{8}$

Correct answer:

$256x^{12}y^{8}$

Explanation:

An exponent outside of a parentheses needs to be distributed to all the numbers and variables in the parentheses. An exponent raised to an exponent should be multiplied.

$\left ( 4x^{3}y^{2} \right )^{4}$

$4^{4}x^{3\cdot 4}y^{2\cdot 4}$

$256x^{12}y^{8}$

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Example Question #25 : Distributing Exponents (Power Rule)

$\left ( 9x^{2}+4y^{3} \right )^{2}$

Possible Answers:

$97x^{4}y^{6}$

$81x^{4}+16y^{5}$

$9x^{4}+4y^{5}$

81x^4+72x^2y^3+16y^6

$9x^{4}+4y^{6}$

Correct answer:

81x^4+72x^2y^3+16y^6

Explanation:

An exponent outside of a parentheses means the entire quantity is being raised to that power. In other words, the quantity inside the parentheses is being multiplied by itself the number of times the outside exponent says.

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

Recall that when like bases are being multiplied together their exponents are added.

$9x^2\cdot 9x^2+9x^2\cdot 4y^3+4y^3\cdot 9x^2+4y^3\cdot 4y^3$

81x^4+36x^2y^3+36x^2y^3+16y^6

81x^4+72x^2y^3+16y^6

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Example Question #21 : Distributing Exponents (Power Rule)

Simplify the expression $\left ( 2x^3\right )^5$

Possible Answers:

32x^8

2x^8

$32x^{15}$

$2x^{15}$

$32768x^{15}$

Correct answer:

$32x^{15}$

Explanation:

$\left ( 2x^5\right )^3=\left ( 2x^5\right )\left ( 2x^5\right )\left ( 2x^5\right )$

$8x^{15}$

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Example Question #21 : Distributing Exponents (Power Rule)

Simplify the expression: $\left ( \frac{x^2y^3z^{-2}}{x^{-2}z^4}\right )^{-3}$

Possible Answers:

$\frac{x^{12}y^{9}}{z^{18}}$

$\frac{z^{18}}{x^{12}y^{9}}$

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

$\frac{z^6}{y^9}$

Correct answer:

$\frac{z^{18}}{x^{12}y^{9}}$

Explanation:

First simplify the expression inside the parentheses.

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

$\left ( \frac{x^4y^3}{z^6}\right )^{-3}$

Then distribute the exponent.

$\frac{x^{-12}y^{-9}}{z^{-18}}$

Rearrange the expression so that there are no more negative exponents.

$\frac{z^{18}}{x^{12}y^{9}}$

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Example Question #22 : Distributing Exponents (Power Rule)

Simplify:

(10^8)^7

Possible Answers:

10^9

8*10^7

7*10^8

$10^{15}$

$10^{56}$

Correct answer:

$10^{56}$

Explanation:

When exponents are being raised by another exponent, we just multiply the powers.

$(10^8)^7=10^{8*7}=10^{56}$

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Example Question #21 : Distributing Exponents (Power Rule)

Simplify:

$(-16^{-5})^{-9}$

Possible Answers:

$16^{45}$

$16^{14}$

$-16^{14}$

$-16^{45}$

$-16^{-14}$

Correct answer:

$-16^{45}$

Explanation:

When exponents are being raised by another exponent, we just multiply the powers.

$(-16^{-5})^{-9}=(-16)^{-5*-9}=-16^{45}$

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Example Question #21 : Distributing Exponents (Power Rule)

Simplify:

((3^5)^7)^8

Possible Answers:

$3^{20}$

$3^{280}$

$3^{144}$

$3^{43}$

$3^{288}$

Correct answer:

$3^{280}$

Explanation:

When exponents are being raised by another exponent, we just multiply the powers.

$((3^5)^7)^8=3^{5*7*8}=3^{280}$

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Example Question #31 : Distributing Exponents (Power Rule)

Simplify:

$(24^{12})^{\frac{1}{18}}$

Possible Answers:

$9\sqrt[3]{4}$

$4\sqrt[3]{9}$

$24\sqrt[3]{2}$

Correct answer:

$4\sqrt[3]{9}$

Explanation:

When exponents are being raised by another exponent, we just multiply the powers.

$(24^{12})^{\frac{1}{18}}=24^{12*\frac{1}{18}}=24^{\frac{2}{3}}$

When we have fractional exponents, we convert like this:

$x^{\frac{a}b{}}=\sqrt[b]{x^a}$ . is the index of the radical which is the denominator of the fractional exponent, is the power that will raise the base which is the numerator of the fractional exponent and is the base.

$24^{\frac{2}{3}}=\sqrt[3]{24^2}=\sqrt[3]{576}$ The perfect cube we can get is .

$\sqrt[3]{576}=\sqrt[3]{64*9}=4\sqrt[3]{9}$

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Example Question #31 : Distributing Exponents (Power Rule)

Evaluate:

(2^9)^4

Possible Answers:

$2^{13}$

4^8

$2^{36}$

$4^{10}$

8^9

Correct answer:

$2^{36}$

Explanation:

When dealing exponents being raised by a power, we multiply the exponents and keep the base.

$(2^9)^4=2^{9*4}=2^{36}$

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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 #21 : Distributing Exponents (Power Rule)

Example Question #24 : Distributing Exponents (Power Rule)

Example Question #25 : Distributing Exponents (Power Rule)

Example Question #21 : Distributing Exponents (Power Rule)

Example Question #21 : Distributing Exponents (Power Rule)

Example Question #22 : Distributing Exponents (Power Rule)

Example Question #21 : Distributing Exponents (Power Rule)

Example Question #21 : Distributing Exponents (Power Rule)

Example Question #31 : Distributing Exponents (Power Rule)

Example Question #31 : Distributing Exponents (Power Rule)

All Algebra II Resources

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