How do you simplify ${(\frac{216}{729})}^{\frac{2}{3}}$ $(216/729)^(2/3)$ ?

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Answer 1 · 2016-09-01T22:08:15

$\frac{4}{9}$ $4/9$

It is a real advantage to know all the powers up to 1000 by heart.

We would then know at once that $216 = 6^{3} and 729 = 9^{3}$ $" "216 = 6^3 and 729 = 9^3$

Else you have to find the prime factors first.

An index that is a fraction shows both a root and a power.

$x^{\frac{p}{q}} = \sqrt[q]{x^{p}} = {(\sqrt[q]{x})}^{p}$ $x^(p/q) = rootq(x^p) = (rootq x)^p$

${(\frac{216}{729})}^{\frac{2}{3}} = {(\sqrt[3]{\frac{216}{729}})}^{2}$ $(216/729)^(2/3) = (root3 (216/729))^2$

= ${(\frac{6}{9})}^{2} = {(\frac{2}{3})}^{2}$ $(6/9)^2 " = "(2/3)^2$

= $\frac{4}{9}$ $4/9$

How do you simplify (216/729)^(2/3)(216729)23(216/729)^(2/3)?