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.
Start Solution
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 2
Now, 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.