Paul's Online Notes
Home / Algebra / Preliminaries / Rational Exponents
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

### Section 1.2 : Rational Exponents

6. Evaluate the following expression and write the answer as a single number without exponents.

${\left( {\frac{8}{{343}}} \right)^{ - \frac{2}{3}}}$

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,

${\left( {\frac{8}{{343}}} \right)^{ - \frac{2}{3}}} = {\left( {\frac{{343}}{8}} \right)^{\,\frac{2}{3}}} = \frac{{{{343}^{\frac{2}{3}}}}}{{{8^{\frac{2}{3}}}}} = \frac{{{{\left( {{{343}^{\frac{1}{3}}}} \right)}^2}}}{{{{\left( {{8^{\frac{1}{3}}}} \right)}^2}}}$ Show Step 2

Now, recalling how the first two practice problems worked we can see that,

${343^{\frac{1}{3}}} = 7\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}{8^{\frac{1}{3}}} = 2$

Therefore,

${\left( {\frac{8}{{343}}} \right)^{ - \frac{2}{3}}} = {\left( {\frac{{343}}{8}} \right)^{\,\frac{2}{3}}} = \frac{{{{343}^{\frac{2}{3}}}}}{{{8^{\frac{2}{3}}}}} = \frac{{{{\left( {{{343}^{\frac{1}{3}}}} \right)}^2}}}{{{{\left( {{8^{\frac{1}{3}}}} \right)}^2}}} = \frac{{{7^2}}}{{{2^2}}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{49}}{4}}}$