How do you evaluate and simplify $\frac{64^{\frac{5}{9}} \cdot 64^{\frac{2}{9}}}{4^{\frac{3}{4}}}$ $(64^(5/9)*64^(2/9))/4^(3/4)$ ?

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How do you evaluate and simplify $\frac{64^{\frac{5}{9}} \cdot 64^{\frac{2}{9}}}{4^{\frac{3}{4}}}$ $(64^(5/9)*64^(2/9))/4^(3/4)$ ?

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Answer 1 · 2018-05-09T20:37:10

the answer
$4^{\frac{84}{36} - \frac{27}{36}} = 4^{\frac{57}{36}} = 4^{\frac{19}{12}}$ $4^(84/36-27/36)=4^(57/36)=4^(19/12)$

$= 8.9797$ $=8.9797$

Explanation:

show below

$\frac{64^{\frac{5}{9}} \cdot 64^{\frac{2}{9}}}{4^{\frac{3}{4}}}$ $(64^(5/9)*64^(2/9))/4^(3/4)$

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

$64 = 4^{3}$ $64=4^3$

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

$4^{\frac{21}{9}} \cdot 4^{- \frac{3}{4}} = 4^{\frac{21}{9} - \frac{3}{4}}$ $4^(21/9)*4^(-3/4)=4^(21/9-3/4)$

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

$= 4^{\frac{19}{12}} = 8.9797$ $=4^(19/12)=8.9797$

The index answer is probably better.

Answer 2 · 2018-05-09T21:07:05

$2^{\frac{19}{6}}$ $2^(19/6)$

Explanation:

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

$x^{a} \div x^{b} = x^{a - b}$ $x^a -: x^b=x^(a-b)$

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

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

using the third law
$64 = 2^{6}$ $64=2^6$ so $64^{\frac{7}{9}} = {(2^{6})}^{\frac{7}{9}} \Rightarrow 2^{\frac{42}{9}} = 2^{\frac{14}{3}}$ $64^(7/9)=(2^6)^(7/9)=>2^(42/9)=2^(14/3)$

$4^{\frac{3}{4}} = {(2^{2})}^{\frac{3}{4}} \Rightarrow 2^{\frac{6}{4}} = 2^{\frac{3}{2}}$ $4^(3/4)=(2^2)^(3/4)=>2^(6/4)=2^(3/2)$

Putting these two together

$\Rightarrow \frac{64^{\frac{5}{9}} \times 64^{\frac{2}{9}}}{4^{\frac{3}{4}}} = \frac{2^{\frac{14}{3}}}{2^{\frac{3}{2}}}$ $=> (64^(5/9) xx 64^(2/9))/4^(3/4)=(2^(14/3))/(2^(3/2))$

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

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How do you evaluate and simplify $\frac{64^{\frac{5}{9}} \cdot 64^{\frac{2}{9}}}{4^{\frac{3}{4}}}$ $(64^(5/9)*64^(2/9))/4^(3/4)$ ?

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How do you evaluate and simplify $\frac{64^{\frac{5}{9}} \cdot 64^{\frac{2}{9}}}{4^{\frac{3}{4}}}$ $(64^(5/9)*64^(2/9))/4^(3/4)$ ?