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Algebra II : Simplifying Exponents

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #111 : Multiplying And Dividing Exponents

Simplify $\frac{3x^{4}y^{3}}{(2xy^{7})^{-1}}$ .

Possible Answers:

$6x^{4}y^{21}$

$6x^{5}y^{10}$

$\frac{3x^{5}y^{10}}{2}$

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

$6x^{4}y^{10}$

Correct answer:

$6x^{5}y^{10}$

Explanation:

If a term in the denominator has a negative exponent, it should be moved to the numerator:

$\frac{3x^{4}y^{3}}{(2xy^{7})^{-1}}=(3x^{4}y^{3})(2xy^{7})$

Now multiply like terms, remembering that when multiplying terms with exponents, you add the exponents.

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

$=6x^{4+1}y^{3+7}=6x^{5}y^{10}$

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Example Question #112 : Multiplying And Dividing Exponents

Solve: $\frac{4^{-2}}{6^{-2}}\times \frac{4^{-2}}{3^{-2}}$

Possible Answers:

$\frac{1}{64}$

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

$\frac{1}{81}$

$\frac{64}{81}$

$\frac{81}{64}$

Correct answer:

$\frac{81}{64}$

Explanation:

We can multiply similar based numerators with exponents by simply adding the powers. For the denominators, we will need to convert the negative exponents into fractions since the bases are uncommon.

$\frac{4^{-2}}{6^{-2}}\times \frac{4^{-2}}{3^{-2}} = \frac{4^{-4}}{\frac{1}{6^2}\times \frac{1}{3^2}} = \frac{\frac{1}{4^4}}{\frac{1}{6^2}\times \frac{1}{3^2}}$

Simplify the complex fraction.

$\frac{\frac{1}{4^4}}{\frac{1}{6^2}\times \frac{1}{3^2}} = \frac{\frac{1}{4}\cdot\frac{1}{4}\cdot\frac{1}{4}\cdot\frac{1}{4}}{\frac{1}{36}\times \frac{1}{9}}$

To avoid having to multiply the fractions out, we can simplify the fraction by rewriting the fraction $\frac{1}{36}$ as $(\frac{1}{4}\cdot \frac{1}{9})$ so that we can eliminate common terms in the numerator and denominator.

$\frac{\frac{1}{4}\cdot\frac{1}{4}\cdot\frac{1}{4}\cdot\frac{1}{4}}{(\frac{1}{4}\cdot \frac{1}{9})\times \frac{1}{9}}$

Cancel the common terms and simplify the fractions.

$\frac{\frac{1}{4}\cdot\frac{1}{4}\cdot\frac{1}{4}}{\frac{1}{9}\times \frac{1}{9}} = \frac{\frac{1}{64}}{\frac{1}{81}}$

Rewrite the complex fraction using a division sign.

$\frac{\frac{1}{64}}{\frac{1}{81}} = \frac{1}{64} \div \frac{1}{81}$

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

$\frac{1}{64} \times 81 = \frac{81}{64}$

The answer is: $\frac{81}{64}$

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Example Question #113 : Multiplying And Dividing Exponents

Simplify: $\frac{x^4y^3b^2}{x^2y^3b^3}$

Possible Answers:

x^2b

$\frac{b}{x^2}$

$\frac{x^2}{b}$

$\frac{b^2}{x}$

$\frac{x}{b}$

Correct answer:

$\frac{x^2}{b}$

Explanation:

$\frac{x^4y^3b^2}{x^2y^3b^3}$

Subtract the exponents of like terms since $\frac{a^m}{a^n}=a^{m-n}$

$x^2b^{-1}$

$\boldsymbol{\frac{x^2}{b}}$ since $a^{-m}=\frac{1}{a^m}$ .

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Example Question #111 : Simplifying Exponents

Simplify: 23^7*23^8

Possible Answers:

$23^{9}$

$23^{56}$

$23^{15}$

$23^{19}$

$23^{10}$

Correct answer:

$23^{15}$

Explanation:

When multiplying exponents with the same base, we add the exponents and keep the base the same.

$23^7*23^8=23^{7+8}=23^{15}$

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Example Question #112 : Simplifying Exponents

Simplify: $2^4*2^{-12}$

Possible Answers:

2^8

$2^{-48}$

$2^{-8}$

$2^{16}$

$2^{-16}$

Correct answer:

$2^{-8}$

Explanation:

When multiplying exponents with the same base, we add the exponents and keep the base the same.

$2^4*2^{-12}=2^{4+(-12)}=2^{-8}$

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Example Question #113 : Simplifying Exponents

Simplify: $(-3)^7*(-3)^{-2}$

Possible Answers:

-3^5

9^5

3^9

$9^{-5}$

$-3^{14}$

Correct answer:

-3^5

Explanation:

When multiplying exponents with the same base, we add the exponents and keep the base the same.

$(-3)^7*(-3)^{-2}=(-3)^{7+(-2)}=(-3)^{5}$

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Example Question #117 : Multiplying And Dividing Exponents

Simplify: 9^4*3^8

Possible Answers:

$3^{16}$

$27^{18}$

$3^{9}$

$27^{6}$

$3^{10}$

Correct answer:

$3^{16}$

Explanation:

Although we have different bases, we know that 9=3^2 . That means 9^4=(3^2)^4=3^8 When multiplying exponents with the same base, we add the exponents and keep the base the same.

$3^8*3^8=3^{8+8}=3^{16}$

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Example Question #114 : Simplifying Exponents

Simplify: $7^{90}\div7^{15}$

Possible Answers:

$7^{75}$

$7^{105}$

7^6

$7^{18}$

$7^{65}$

Correct answer:

$7^{75}$

Explanation:

When dividing exponents with the same base, we subtract the exponents while keeping the base the same.

$7^{90}\div7^{15}=7^{90-15}=7^{75}$

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Example Question #115 : Simplifying Exponents

Simplify: $12^{9}\div12^{12}$

Possible Answers:

12^3

$12^{108}$

144^3

$12^{-3}$

$144^{-3}$

Correct answer:

$12^{-3}$

Explanation:

When dividing exponents with the same base, we subtract the exponents while keeping the base the same.

$12^{9}\div12^{12}=12^{9-12}=12^{-3}$

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Example Question #116 : Multiplying And Dividing Exponents

Simplify:

$\frac{16x^4y^6z^{10}}{40x^8y^2z^5}$

Possible Answers:

$\frac{2y^4z^5}{16x^4}$

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

$\frac{2y^3z^5}{5x^4}$

$\frac{2y^4z^5}{5x^2}$

$\frac{2y^4z^5}{5x^4}$

Correct answer:

$\frac{2y^4z^5}{5x^4}$

Explanation:

Remember, when simplifying exponents while dividing, subtract exponents. Simplify the like terms separately:

$\frac{16}{40}=\frac{2}{5}$ ,

$\frac{x^4}{x^8}=\frac{1}{x^4}$ ,

$\frac{y^6}{y^2}=y^4$ ,

$\frac{z^{10}}{z^5}=z^5$ .

Then, put all of those together to get your final answer:

$\frac{2y^4z^5}{5x^4}$ .

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Algebra II : Simplifying Exponents

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #111 : Multiplying And Dividing Exponents

Example Question #112 : Multiplying And Dividing Exponents

Example Question #113 : Multiplying And Dividing Exponents

Example Question #111 : Simplifying Exponents

Example Question #112 : Simplifying Exponents

Example Question #113 : Simplifying Exponents

Example Question #117 : Multiplying And Dividing Exponents

Example Question #114 : Simplifying Exponents

Example Question #115 : Simplifying Exponents

Example Question #116 : Multiplying And Dividing Exponents

All Algebra II Resources

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