Section 1.2 : Rational Exponents
4. Evaluate the following expression and write the answer as a single number without exponents.
\[{27^{ - \frac{5}{3}}}\]Show All Steps Hide All Steps
Hint : Don’t forget your basic exponent rules and how the first two practice problems worked.
Let’s first recall our basic exponent rules and note that we can easily write this as,
\[{27^{ - \,\,\frac{5}{3}}} = \frac{1}{{{{27}^{\frac{5}{3}}}}} = \frac{1}{{{{\left( {{{27}^{\frac{1}{3}}}} \right)}^5}}}\] Show Step 2Now, recalling how the first two practice problems worked we can see that,
\[{27^{\frac{1}{3}}} = 3\]because \({3^3} = 27\).
Therefore,
\[{27^{ - \,\,\frac{5}{3}}} = \frac{1}{{{{27}^{\frac{5}{3}}}}} = \frac{1}{{{{\left( {{{27}^{\frac{1}{3}}}} \right)}^5}}} = \frac{1}{{{{\left( 3 \right)}^5}}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{1}{{243}}}}\]Sometimes the easiest way to do these kinds of problems when you first run into them is to break them up into manageable steps as we did here.