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Algebra II : Simplifying Exponents

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Example Questions

Example Question #1 : Simplifying Exponents

Simplify.

$x^{-6}x^3x^9$ $\displaystyle x^{-6}x^3x^9$

Possible Answers:

$x^{-12}$ $\displaystyle x^{-12}$

$\displaystyle x$

$x^{12}$ $\displaystyle x^{12}$

x^6 $\displaystyle x^6$

$x^{-6}$ $\displaystyle x^{-6}$

Correct answer:

x^6 $\displaystyle x^6$

Explanation:

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

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Example Question #3461 : Algebra Ii

Evaluate the following expression

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

Possible Answers:

$\frac{1}{25}$ $\displaystyle \frac{1}{25}$

$\displaystyle 1$

$5^{20}$ $\displaystyle 5^{20}$

$\frac{1}{2}$ $\displaystyle \frac{1}{2}$

$\displaystyle 25$

Correct answer:

$\displaystyle 1$

Explanation:

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

We can also solve this problem using a different apporach

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

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

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Example Question #1 : Simplifying Exponents

Simplify the following expression

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

Possible Answers:

$\displaystyle yz$

$y^{\frac{2}{3}}z^{\frac{2}{3}}$ $\displaystyle y^{\frac{2}{3}}z^{\frac{2}{3}}$

$y^{6}z^{6}$ $\displaystyle y^{6}z^{6}$

$y^{2}z^{2}$ $\displaystyle y^{2}z^{2}$

Correct answer:

$y^{6}z^{6}$ $\displaystyle y^{6}z^{6}$

Explanation:

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

Alternatively,

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

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Example Question #801 : Mathematical Relationships And Basic Graphs

Simplify the following expression

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

Possible Answers:

36^3 $\displaystyle 36^3$

6^5 $\displaystyle 6^5$

6^6 $\displaystyle 6^6$

$\displaystyle 1$

Correct answer:

$\displaystyle 1$

Explanation:

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

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

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Example Question #1 : Simplifying Exponents

Evaluate the following expression

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

Possible Answers:

100,000 $\displaystyle 100,000$

$\displaystyle .01$

$\displaystyle .0001$

$\displaystyle 1,000,000$

$\displaystyle 10$

Correct answer:

$\displaystyle 1,000,000$

Explanation:

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

Alternatively,

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

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Example Question #1 : Simplifying Exponents

Simplify:

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

Possible Answers:

$\frac{4a^{2}b^{2}}{12}$ $\displaystyle \frac{4a^{2}b^{2}}{12}$

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

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

$3a^{12}b^{-8}$ $\displaystyle 3a^{12}b^{-8}$

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

Correct answer:

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

Explanation:

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

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

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

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

Step 2: Reduce the fraction

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

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Example Question #2 : Simplifying Exponents

Simplify:

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

Possible Answers:

$q^{9}r^{8}s^{17}$ $\displaystyle q^{9}r^{8}s^{17}$

$q^{5}r^{2}s^{15}$ $\displaystyle q^{5}r^{2}s^{15}$

$q^{14}r^{15}s^{16}$ $\displaystyle q^{14}r^{15}s^{16}$

qrs $\displaystyle qrs$

$q^{3.5}r^{1.3}s^{16}$ $\displaystyle q^{3.5}r^{1.3}s^{16}$

Correct answer:

$q^{5}r^{2}s^{15}$ $\displaystyle q^{5}r^{2}s^{15}$

Explanation:

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

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Example Question #4 : Multiplying And Dividing Exponents

Rewrite using a single exponent.

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

Possible Answers:

$x^{5}$ $\displaystyle x^{5}$

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

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

$x^{11}$ $\displaystyle x^{11}$

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

Correct answer:

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

Explanation:

Based on the property for dividing exponents:

$\frac{a^{m}}{a^{n}}=a^{m-n}$ $\displaystyle \frac{a^{m}}{a^{n}}=a^{m-n}$

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

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

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Example Question #1 : Simplifying Exponents

Simplify:

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

Possible Answers:

$\frac{bc^3}{2a^{12}}$ $\displaystyle \frac{bc^3}{2a^{12}}$

$\frac{bc^3}{98a^{12}}$ $\displaystyle \frac{bc^3}{98a^{12}}$

$\frac{a^{12}}{2bc^3}$ $\displaystyle \frac{a^{12}}{2bc^3}$

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

$\frac{2}{a^{2}bc^3}$ $\displaystyle \frac{2}{a^{2}bc^3}$

Correct answer:

$\frac{bc^3}{2a^{12}}$ $\displaystyle \frac{bc^3}{2a^{12}}$

Explanation:

Simplify:

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

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

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

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

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

Step 3: Simplify

$\frac{bc^3}{2a^{12}}$ $\displaystyle \frac{bc^3}{2a^{12}}$

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Example Question #2 : Multiplying And Dividing Exponents

Simplify:

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

Possible Answers:

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

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

$\frac{x}{yz^2}$ $\displaystyle \frac{x}{yz^2}$

$\frac{x^{10}}{zy^{12}}$ $\displaystyle \frac{x^{10}}{zy^{12}}$

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

Correct answer:

$\frac{x^{10}}{zy^{12}}$ $\displaystyle \frac{x^{10}}{zy^{12}}$

Explanation:

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

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

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

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

${\color{Red} \frac{x^{10}}{zy^{12}}}$

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Algebra II : Simplifying Exponents

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #1 : Simplifying Exponents

Example Question #3461 : Algebra Ii

Example Question #1 : Simplifying Exponents

Example Question #801 : Mathematical Relationships And Basic Graphs

Example Question #1 : Simplifying Exponents

Example Question #1 : Simplifying Exponents

Example Question #2 : Simplifying Exponents

Example Question #4 : Multiplying And Dividing Exponents

Example Question #1 : Simplifying Exponents

Example Question #2 : Multiplying And Dividing Exponents

All Algebra II Resources

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