Algebra II : Multiplying and Dividing Exponents

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #1 : Multiplying And Dividing Exponents

Simplify.

\(\displaystyle x^{-6}x^3x^9\)

Possible Answers:

\(\displaystyle x^{-6}\)

\(\displaystyle x^{12}\)

\(\displaystyle x^{-12}\)

\(\displaystyle x\)

\(\displaystyle x^6\)

Correct answer:

\(\displaystyle x^6\)

Explanation:

Put the negative exponent on the bottom so that you have \(\displaystyle \frac{x^{12}}{x^{6}}\) which simplifies further to \(\displaystyle x^{6}\).

Example Question #1 : Simplifying Exponents

Evaluate the following expression

\(\displaystyle (5^{0})^{2}\)

Possible Answers:

\(\displaystyle 1\)

\(\displaystyle \frac{1}{2}\)

\(\displaystyle 5^{20}\)

\(\displaystyle \frac{1}{25}\)

\(\displaystyle 25\)

Correct answer:

\(\displaystyle 1\)

Explanation:

\(\displaystyle (5^{0})^{2}=(1)^{2}=1\)

We can also solve this problem using a different apporach

\(\displaystyle (5^{0})^{2}=5^{0\times2}=5^{0}=1\)

Remember that any number raised to the 0th power equals 1

Example Question #1 : Simplifying Exponents

Simplify the following expression

\(\displaystyle ((yz)^2)^{3}\)

Possible Answers:

\(\displaystyle y^{6}z^{6}\)

\(\displaystyle yz\)

\(\displaystyle y^{\frac{2}{3}}z^{\frac{2}{3}}\)

\(\displaystyle y^{2}z^{2}\)

Correct answer:

\(\displaystyle y^{6}z^{6}\)

Explanation:

\(\displaystyle ((yz)^{2})^{3}=(yz)^{6}=y^{6}z^{6}\)

Alternatively,

\(\displaystyle ((yz)^{2})^{3}=(y^{2}z^{2})^3=y^{2\times3}z^{2\times3}=y^{6}z^{6}\)

Example Question #1 : Multiplying And Dividing Exponents

Simplify the following expression

\(\displaystyle ((6^2)^{0})^{3}\)

Possible Answers:

\(\displaystyle 6^5\)

\(\displaystyle 1\)

\(\displaystyle 36^3\)

\(\displaystyle 6^6\)

Correct answer:

\(\displaystyle 1\)

Explanation:

\(\displaystyle ((6^2)^{0})^{3}=6^{2\times 0\times 3}=6^0=1\)

Remember that any number raised to the 0th power equals 1

Example Question #4 : Simplifying Exponents

Evaluate the following expression

\(\displaystyle \left (\frac{10^{3}}{10^{0}} \right )^{2}\)

Possible Answers:

\(\displaystyle 1,000,000\)

\(\displaystyle 100,000\)

\(\displaystyle .0001\)

\(\displaystyle .01\)

\(\displaystyle 10\)

Correct answer:

\(\displaystyle 1,000,000\)

Explanation:

\(\displaystyle \left (\frac{10^{3}}{10^{0}} \right )^{2}=\frac{10^{3\times2}}{10^{0\times2}}=\frac{10^{6}}{10^{0}}=\frac{1000000}{1}=1,000,000\)

Alternatively,

\(\displaystyle \left (\frac{10^{3}}{10^{0}} \right )^{2}=\left (10^{3-0} \right )^{2}=\left (10^{3} \right )^{2}=10^{3\times 2}=10^{6}=1,000,000\)

Example Question #1 : Simplifying Exponents

Simplify:

\(\displaystyle \frac{4a^{-5}b^{3}c^{2}}{12a^{-7}b^{-5}c^{-2}}\)

Possible Answers:

\(\displaystyle \frac{1}{3a^{-12}b^{-2}}\)

\(\displaystyle 3a^{12}b^{-8}\)

\(\displaystyle \frac{a^{-12}b^{-2}c^{0}}{3}\)

\(\displaystyle \frac{4a^{2}b^{2}}{12}\)

\(\displaystyle \frac{a^{2}b^{8}c^{4}}{3}\)

Correct answer:

\(\displaystyle \frac{a^{2}b^{8}c^{4}}{3}\)

Explanation:

\(\displaystyle \frac{4a^{-5}b^{3}c^{2}}{12a^{-7}b^{-5}c^{-2}}\)

Step 1: Simplify the exponents using the division of exponents rule (subtract exponents in demoninator from exponents in numerator).

\(\displaystyle \frac{4a^{-5-(-7)}b^{3-(-5)}c^{2-(-2)}}{12}\)

\(\displaystyle \frac{4a^{2}b^{8}c^{4}}{12}\)

Step 2: Reduce the fraction

\(\displaystyle \frac{a^{2}b^{8}c^{4}}{3}\)

Example Question #2 : Simplifying Exponents

Simplify:

\(\displaystyle \frac{q^{7}r^{5}s^{16}}{q^{2}r^{3}s}\)

Possible Answers:

\(\displaystyle qrs\)

\(\displaystyle q^{9}r^{8}s^{17}\)

\(\displaystyle q^{5}r^{2}s^{15}\)

\(\displaystyle q^{3.5}r^{1.3}s^{16}\)

\(\displaystyle q^{14}r^{15}s^{16}\)

Correct answer:

\(\displaystyle q^{5}r^{2}s^{15}\)

Explanation:

Follow the division of exponents rule.  Subract the exponents in the denominator from the exponents in the numerator.

Example Question #4 : Multiplying And Dividing Exponents

Rewrite using a single exponent.

\(\displaystyle \frac{(6x)^{8}}{(6x)^{3}}\)

Possible Answers:

\(\displaystyle x^{5}\)

\(\displaystyle (6x)^{24}\)

\(\displaystyle (6x)^{11}\)

\(\displaystyle x^{11}\)

\(\displaystyle (6x)^{5}\)

Correct answer:

\(\displaystyle (6x)^{5}\)

Explanation:

Based on the property for dividing exponents:

\(\displaystyle \frac{a^{m}}{a^{n}}=a^{m-n}\)

In this problem, a is equal to \(\displaystyle (6x)\), so

\(\displaystyle \frac{(6x)^{8}}{(6x)^{3}}=(6x)^{8-3}=(6x)^{5}\)

Example Question #1 : Simplifying Exponents

Simplify:

\(\displaystyle \frac{7a^{-7}b^{-8}c^{4}}{14a^{5}b^{-9}c{}}\)

Possible Answers:

\(\displaystyle \frac{bc^3}{2a^{12}}\)

\(\displaystyle \frac{bc^3}{98a^{12}}\)

\(\displaystyle \frac{a^{12}}{2bc^3}\)

\(\displaystyle \frac{7bc^3}{14a^2}\)

\(\displaystyle \frac{2}{a^{2}bc^3}\)

Correct answer:

\(\displaystyle \frac{bc^3}{2a^{12}}\)

Explanation:

Simplify:

\(\displaystyle \frac{7a^{-7}b^{-8}c^{4}}{14a^{5}b^{-9}c{}}\)

Step 1: Use the division of exponents rule, and subtract the exponents in the denominator from the exponents in the numerator

\(\displaystyle \frac{7a^{-12}bc^3}{14}\)

Step 2: Move negative exponents in the numerator to the denominator

\(\displaystyle \frac{7bc^3}{14a^{12}}\)

Step 3: Simplify

\(\displaystyle \frac{bc^3}{2a^{12}}\)

Example Question #3 : Simplifying Exponents

Simplify:

\(\displaystyle \frac{x^{6}zy^{-9}}{x^{-4}z^2y^3}\)

Possible Answers:

\(\displaystyle \frac{x^{10}z^3}{y^6}\)

\(\displaystyle x^2z^{-1}y^6\)

\(\displaystyle \frac{y^6}{x^{10}z^3}\)

\(\displaystyle \frac{x}{yz^2}\)

\(\displaystyle \frac{x^{10}}{zy^{12}}\)

Correct answer:

\(\displaystyle \frac{x^{10}}{zy^{12}}\)

Explanation:

Step 1: Use the division of exponents rule.  Subtract the exponents in the numerator from the exponents in the denominator.

\(\displaystyle {x^{6+4}z^{1-2}y^{-9-3}\)

\(\displaystyle x^{10}z^{-1}y^{-12}\)

Step 2: Represent the negative exponents as positive ones by moving them to the denominator:

\(\displaystyle {\color{Red} \frac{x^{10}}{zy^{12}}}\)

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