# How do you evaluate and simplify (64^(5/9)*64^(2/9))/4^(3/4)?

May 9, 2018

${4}^{\frac{84}{36} - \frac{27}{36}} = {4}^{\frac{57}{36}} = {4}^{\frac{19}{12}}$

$= 8.9797$

#### Explanation:

show below

$\frac{{64}^{\frac{5}{9}} \cdot {64}^{\frac{2}{9}}}{4} ^ \left(\frac{3}{4}\right)$

${\left(64\right)}^{\frac{5}{9} + \frac{2}{9}} / {4}^{\frac{3}{4}} = {\left(64\right)}^{\frac{7}{9}} / {4}^{\frac{3}{4}}$

$64 = {4}^{3}$

${\left({4}^{3}\right)}^{\frac{7}{9}} / {4}^{\frac{3}{4}} = {4}^{\frac{21}{9}} / {4}^{\frac{3}{4}}$

${4}^{\frac{21}{9}} \cdot {4}^{- \frac{3}{4}} = {4}^{\frac{21}{9} - \frac{3}{4}}$

${4}^{\frac{84}{36} - \frac{27}{36}} = {4}^{\frac{57}{36}} = {4}^{\frac{19}{12}}$

$= {4}^{\frac{19}{12}} = 8.9797$

The index answer is probably better.

May 9, 2018

${2}^{\frac{19}{6}}$

#### Explanation:

Here are some laws of indices
${x}^{a} \times {x}^{b} = {x}^{a + b}$

${x}^{a} \div {x}^{b} = {x}^{a - b}$

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

Using the first law
${64}^{\frac{5}{9}} \times {64}^{\frac{2}{9}} = {64}^{\frac{7}{9}}$

using the third law
$64 = {2}^{6}$ so ${64}^{\frac{7}{9}} = {\left({2}^{6}\right)}^{\frac{7}{9}} \implies {2}^{\frac{42}{9}} = {2}^{\frac{14}{3}}$

${4}^{\frac{3}{4}} = {\left({2}^{2}\right)}^{\frac{3}{4}} \implies {2}^{\frac{6}{4}} = {2}^{\frac{3}{2}}$

Putting these two together

$\implies \frac{{64}^{\frac{5}{9}} \times {64}^{\frac{2}{9}}}{4} ^ \left(\frac{3}{4}\right) = \frac{{2}^{\frac{14}{3}}}{{2}^{\frac{3}{2}}}$

Using the second law
$\frac{{2}^{\frac{14}{3}}}{{2}^{\frac{3}{2}}} = {2}^{\frac{14}{3} - \frac{3}{2}} = {2}^{\frac{28}{6} - \frac{9}{6}} = {2}^{\frac{19}{6}}$