Algebra II : Exponents

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #108 : Simplifying Exponents

Simplify: \displaystyle 21^{24}\div21^{-19}

Possible Answers:

\displaystyle 21^{14}

\displaystyle 21^{34}

\displaystyle 21^{43}

\displaystyle 21^5

\displaystyle 21^4

Correct answer:

\displaystyle 21^{43}

Explanation:

When dividing exponents with the same base, we subtract the exponents and keep the base the same.

\displaystyle 21^{24}\div21^{-19}=21^{24-(-19)}=21^{43}

Example Question #111 : Multiplying And Dividing Exponents

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

Possible Answers:

\displaystyle 6x^{5}y^{10}

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

\displaystyle 6x^{4}y^{10}

\displaystyle 6x^{4}y^{21}

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

Correct answer:

\displaystyle 6x^{5}y^{10}

Explanation:

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

\displaystyle \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.

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

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

Example Question #112 : Multiplying And Dividing Exponents

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

Possible Answers:

\displaystyle \frac{1}{64}

\displaystyle \frac{1}{81}

\displaystyle \frac{64}{81}

\displaystyle \frac{81}{64}

Correct answer:

\displaystyle \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.

\displaystyle \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.

\displaystyle \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 \displaystyle \frac{1}{36} as \displaystyle (\frac{1}{4}\cdot \frac{1}{9}) so that we can eliminate common terms in the numerator and denominator.

\displaystyle \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.

\displaystyle \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.

\displaystyle \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.

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

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

Example Question #441 : Exponents

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

Possible Answers:

\displaystyle x^2b

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

\displaystyle \frac{x}{b}

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

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

Correct answer:

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

Explanation:

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

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

\displaystyle x^2b^{-1}

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

Example Question #442 : Exponents

Simplify: \displaystyle 23^7*23^8

Possible Answers:

\displaystyle 23^{9}

\displaystyle 23^{10}

\displaystyle 23^{15}

\displaystyle 23^{56}

\displaystyle 23^{19}

Correct answer:

\displaystyle 23^{15}

Explanation:

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

\displaystyle 23^7*23^8=23^{7+8}=23^{15}

Example Question #443 : Exponents

Simplify: \displaystyle 2^4*2^{-12}

Possible Answers:

\displaystyle 2^{-16}

\displaystyle 2^8

\displaystyle 2^{-48}

\displaystyle 2^{-8}

\displaystyle 2^{16}

Correct answer:

\displaystyle 2^{-8}

Explanation:

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

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

Example Question #444 : Exponents

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

Possible Answers:

\displaystyle 9^5

\displaystyle -3^5

\displaystyle -3^{14}

\displaystyle 9^{-5}

\displaystyle 3^9

Correct answer:

\displaystyle -3^5

Explanation:

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

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

Example Question #3577 : Algebra Ii

Simplify: \displaystyle 9^4*3^8

Possible Answers:

\displaystyle 3^{10}

\displaystyle 27^{18}

\displaystyle 27^{6}

\displaystyle 3^{9}

\displaystyle 3^{16}

Correct answer:

\displaystyle 3^{16}

Explanation:

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

\displaystyle 3^8*3^8=3^{8+8}=3^{16}

Example Question #445 : Exponents

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

Possible Answers:

\displaystyle 7^{105}

\displaystyle 7^{18}

\displaystyle 7^{75}

\displaystyle 7^6

\displaystyle 7^{65}

Correct answer:

\displaystyle 7^{75}

Explanation:

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

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

Example Question #446 : Exponents

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

Possible Answers:

\displaystyle 12^{-3}

\displaystyle 12^3

\displaystyle 144^3

\displaystyle 12^{108}

\displaystyle 144^{-3}

Correct answer:

\displaystyle 12^{-3}

Explanation:

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

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

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