# What is the value ofx ifx^(4/5)=(2^8)/(3^8)?

Mar 18, 2017

$x = \frac{1024}{59049}$

#### Explanation:

If $x > 0$ and $a , b$ are any real numbers, then:

${\left({x}^{a}\right)}^{b} = {x}^{a b}$

So we find:

$x = {x}^{1} = {x}^{\frac{4}{5} \cdot \frac{5}{4}} = {\left({x}^{\frac{4}{5}}\right)}^{\frac{5}{4}} = {\left({2}^{8} / {3}^{8}\right)}^{\frac{5}{4}} = {\left({\left(\frac{2}{3}\right)}^{8}\right)}^{\frac{5}{4}} = {\left(\frac{2}{3}\right)}^{8 \cdot \frac{5}{4}} = {\left(\frac{2}{3}\right)}^{10} = {2}^{10} / {3}^{10} = \frac{1024}{59049}$

Mar 18, 2017

$x = {\left(\frac{2}{3}\right)}^{10}$

#### Explanation:

In general if ${x}^{a} = {p}^{b}$ then
$\textcolor{w h i t e}{\text{XXX}} x = {\left({x}^{a}\right)}^{\frac{1}{a}} = {\left({p}^{b}\right)}^{\frac{1}{a}}$

In this case
$\textcolor{w h i t e}{\text{XXX")a = 4/5color(white)("XX")rarrcolor(white)("XX}} \frac{1}{a} = \frac{5}{4}$

$\textcolor{w h i t e}{\text{XXX}} p = \frac{2}{3}$
and
$\textcolor{w h i t e}{\text{XXXXXX}}$after noting that $\frac{{2}^{8}}{{3}^{8}} = {\left(\frac{2}{3}\right)}^{8}$
$\textcolor{w h i t e}{\text{XXX}} b = 8$

So
$\textcolor{w h i t e}{\text{XXX}} x = {\left({\left(\frac{2}{3}\right)}^{8}\right)}^{\frac{5}{4}} = {\left(\frac{2}{3}\right)}^{\frac{8 \cdot 5}{4}} = {\left(\frac{2}{3}\right)}^{10}$

Mar 18, 2017

$x = 0.01734152$

#### Explanation:

${x}^{\frac{4}{5}} = \frac{{2}^{8}}{3} ^ 8$

Make $x$ radical,

$\sqrt[5]{{x}^{4}} = {2}^{8} / {3}^{8}$

Multiply both sides by the index of 5,

${\left(\sqrt[5]{{x}^{4}}\right)}^{5} = {\left({2}^{8} / {3}^{8}\right)}^{5}$
${x}^{4} = {2}^{40} / {3}^{40}$

Root both sides by index of 4,

$\sqrt[4]{{x}^{4}} = \sqrt[4]{{2}^{40} / {3}^{40}}$
${\left({x}^{4}\right)}^{\frac{1}{4}} = {\left({2}^{40} / {3}^{40}\right)}^{\frac{1}{4}}$
$x = {2}^{10} / {3}^{10}$
$x = \frac{1024}{59049}$

Hence $x = 0.01734152$.