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Notes

\({\text{Exponents Properties}}\)
\(\underline{\text{Notes}}\)	\(\underline{\text{Examples}}\)
\(\displaystyle k^ak^b=k^{a+b}\)	\(\displaystyle 7^2\cdot7^3=7^{2+3}=7^5\)
\(\displaystyle \left(k^a \right)^b=k^{a \cdot b}\)	\(\displaystyle \left(7^2 \right)^3=7^{2 \cdot 3}=7^6\)
\(\displaystyle \left(km \right)^a=k^a \cdot m^a\)	\(\displaystyle \left(7x \right)^3=7^3 \cdot x^3=343x^3\)
\(\displaystyle \frac{k^a}{k^b}=k^{a-b}\)	\(\displaystyle \frac{7^5}{7^3}=7^{5-3}=7^2=49\)
\(\displaystyle \left( \frac{k}{m} \right)^a=\frac{k^a}{m^a}\)	\(\displaystyle \left( \frac{2}{3} \right)^5=\frac{2^5}{3^5}\)
\(\displaystyle k^{a/b}=\sqrt[b]{k^a}\)	\(\displaystyle 5^{2/3}=\sqrt[3]{5^2}\)
\(\displaystyle k^{-a}=\frac{1}{k^a}\)	\(\displaystyle 3^{-2}=\frac{1}{3^2}=\frac{1}{9}\)
\(\displaystyle k^0=1\)	\(\displaystyle 3^0=1\)

Problems and Videos

Evaluate

\(\textbf{1)}\) \( \displaystyle 9^{3/2} \)

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The answer is \( 27 \)

\(\textbf{2)}\) \( \displaystyle 8^{1/3} \)

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The answer is \( 2 \)

\(\textbf{3)}\) \( \displaystyle 8^{2/3} \)

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The answer is \( 4 \)

\(\textbf{4)}\) \( \displaystyle 8^{4/3} \)

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The answer is \( 16 \)

\(\textbf{5)}\) \( \displaystyle 8^{5/3} \)

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The answer is \( 32 \)

\(\textbf{6)}\) \( \displaystyle 4^{3/2} \)

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The answer is \( 8 \)

\(\textbf{7)}\) \( \displaystyle 4^{5/2} \)

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The answer is \( 32 \)

\(\textbf{8)}\) \( \displaystyle 16^{1/4} \)

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The answer is \( 2 \)

\(\textbf{9)}\) \( \displaystyle 27^{1/3} \)

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The answer is \( 3 \)

\(\textbf{10)}\) \( \displaystyle 27^{2/3} \)

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The answer is \( 9 \)

\(\textbf{11)}\) \( \displaystyle 125^{2/3} \)

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The answer is \( 25 \)

Challenge Problems

\(\textbf{12)}\) \( \displaystyle 16^{2/5} \)

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The answer is \( 2 \sqrt[5]{8} \, \) or \( \, 2\cdot 2^{3/5} \)

\(\textbf{13)}\) \( (\displaystyle 4^{1/2})( \displaystyle4^{2/3}) \)

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The answer is \( 4\sqrt[3]{2} \, \) or \( \, 4 \cdot 2^{1/3} \)

\(\textbf{14)}\) \( \displaystyle \frac{4}{9}^{-1/2} \)

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The answer is \( \displaystyle \frac{3}{2} \)

\(\textbf{15)}\) \( \displaystyle \frac{1}{x^{2/3}} \)

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The answer is \( \displaystyle \frac{\sqrt[3]{x}}{x} \, \) or \( \, \displaystyle \frac{x^{1/3}}{x} \)

\(\textbf{16)}\) \( \displaystyle \frac{2x^2+3x}{\sqrt[3]{x}} \)

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The answer is \( \displaystyle \frac{2x^{8/3}+3x^{5/3}}{x} \)

\(\textbf{17)}\) \( \displaystyle \frac{zw}{\sqrt[3]{x}} \)

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The answer is \( \displaystyle \frac{x^{2/3}zw}{x} \)

See Related Pages\(\)

\(\bullet\text{ Intro to Exponents}\)
\(\,\,\,\,\,\,\,\,3^2=3 \times 3 =9\)

\(\bullet\text{ Negative Exponents}\)
\(\,\,\,\,\,\,\,\,3^{-2}=\frac{1}{9}\)

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