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Reed College Department of Mathematics & Statistics

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Worksheet Radicals and Rational Exponents

6.

Compute the following. Use "undefined" or "imaginary" for non-real results.
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\(\large{243^{3/5}}\) =
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Hint.
Remember that fractions as exponents can be translated into radical form.
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\(\large x^{m/n} = \left(\sqrt[n]{x}\right)^m\)
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Answer.
\(27\)
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Solution.
Rewrite the fractional exponent as a radical:
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\(\displaystyle 243^{3/5} = \left(\sqrt[5]{243}\right)^{3}\)
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Compute the radical:
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\(\displaystyle \left(\sqrt[5]{243}\right)^{3} = (3)^{3}\)
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Compute the exponent:
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\(\displaystyle (3)^{3} = 27\)
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12.

Simplify the following expression. Write your answer using positive exponents only.
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\(\large{z^{-1/3} z^{2/3}}\) =
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Hint.
Recall the property of exponents:
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\(\displaystyle B^m \cdot B^n = B^{(m+n)}\)
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Remember that negative exponents can be rewritten with positive exponents.
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\(\displaystyle B^{-n} = \frac{1}{B^n}\)
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Answer.
\(z^{\frac{1}{3}}\)
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Solution.
Use the property of exponents:
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\(\displaystyle z^{-1/3} z^{2/3} = z^{((-1/3)+(2/3))}\)
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Off to the side, add the exponents:
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\(\displaystyle \frac{-1}{3} + \frac{2}{3} = \frac{1}{3}\)
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so now we have \(\displaystyle z^{((-1/3)+(2/3))} = z^{1/3}\)
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Simplify your answer:
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\(\displaystyle z^{1/3} = z^{\frac{1}{3}}\)
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