How do you simplify expressions involving variables such as (27p^3q^12)^(1/3)?

1 Answer
Apr 7, 2015

${a}^{\frac{1}{3}}$ is the third root of $x$ or $\sqrt[3]{x}$
so what we want to try to do is express the components of $x$ as powers of $3$ (as much as possible).

Note also that

$\sqrt[n]{s \cdot t} = \sqrt[n]{s} \cdot \sqrt[n]{t}$

For the given example
${\left(27 {p}^{3} {q}^{12}\right)}^{\frac{1}{3}}$

$= {\left({3}^{3} \cdot {p}^{3} \cdot {\left({q}^{4}\right)}^{3}\right)}^{\frac{1}{3}}$

$= \sqrt[3]{{3}^{3}} \cdot \sqrt[3]{{p}^{3}} \cdot \sqrt[3]{{\left({q}^{4}\right)}^{3}}$

$= 3 p {q}^{4}$