# How do you evaluate and simplify 27^(2/3)?

Feb 13, 2017

See the entire solution process below:

#### Explanation:

First, we can use these rule for exponents to modify this expression:

${x}^{\textcolor{red}{a} \times \textcolor{b l u e}{b}} = {\left({x}^{\textcolor{red}{a}}\right)}^{\textcolor{b l u e}{b}}$

${27}^{\frac{2}{3}} = {27}^{\textcolor{red}{\frac{1}{3}} \times \textcolor{b l u e}{2}} = {\left({27}^{\textcolor{red}{\frac{1}{3}}}\right)}^{\textcolor{b l u e}{2}}$

We can now use rule of roots/exponents to modify the expression within parenthesis:

${x}^{\frac{1}{\textcolor{red}{n}}} = \sqrt[\textcolor{red}{n}]{x}$

${\left({27}^{\frac{1}{\textcolor{red}{3}}}\right)}^{2} = {\left(\sqrt[\textcolor{red}{3}]{27}\right)}^{2} = {\left(3\right)}^{2} = \left(3 \times 3\right) = 9$

Feb 13, 2017

You can also use logarithmic functions, if you have a logarithm table or at least a calculator that can do log functions, if not complex exponential ones.

#### Explanation:

Using the transformation of ${27}^{\frac{2}{3}} = \left(\frac{2}{3}\right) \cdot \log 27$
we calculate
$\left(\frac{2}{3}\right) \cdot 1.431 = 0.9542$
Taking the inverse (anti-log, or exponent) of this value gives us the answer:

${10}^{0.9542} = 9$