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Section 1.2 : Rational Exponents
2. Evaluate the following expression and write the answer as a single number without exponents.
\[{\left( { - 125} \right)^{\frac{1}{3}}}\]
Hint : Recall that \({b^{\frac{1}{n}}}\) is really asking what number did we raise to the \(n\) to get \(b\). Or in other words,
\[{b^{\frac{1}{n}}} = ?\hspace{0.25in}\hspace{0.25in}{\mbox{is equivalent to}}\hspace{0.25in}{?^n} = b\]
For this problem we know that \({5^3} = 125\). Therefore, we also know that \({\left( { - 5} \right)^3} = - 125\) and so we further know that,
\[{\left( { - 125} \right)^{\frac{1}{3}}} = \require{bbox} \bbox[2pt,border:1px solid black]{{ - 5}}\]Note that if you aren’t sure of the answer to these kinds of problems all you really need to do is set up
\[{?^3} = - 125\]and start trying integers until you get the one you need. We also know that because the result is a negative number we had to have a negative number to start off with since we can’t turn a positive number into a negative number simply by raising it to an integer.