How do you simplify $\frac{{(3 x^{7} y^{3})}^{4}}{12 x y^{3}}$ $(3x^7y^3)^4/(12xy^3)$ ?

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Answer 1 · 2015-06-12T16:56:01

$\frac{27}{4} x^{27} . y^{9}$ $color(blue)(27/4x^27.y^9$

$\frac{{(3 x^{7} y^{3})}^{4}}{12 x y^{3}}$ $(3x^7y^3)^4/(12xy^3)$

${(3 x^{7} y^{3})}^{4} = 3^{4} . x^{7.4} . y^{3.4}$ $(3x^7y^3)^color(red)(4) = 3^color(red)(4).x^(7 . color(red)(4)). y^(3 .color(red)(4)$

$= 3^{4} x^{28} y^{12}$ $= 3^4x^28y^12$

now the expression can be re written as :
$\frac{3^{4} x^{28} y^{12}}{12 x y^{3}}$ $(3^4x^28y^12)/(12xy^3)$

Note :
$a^{m} . a^{n} = a^{m + n}$ $color(blue)(a^m . a^n = a^(m+n)$
$\frac{a^{m}}{a^{n}} = a^{m - n}$ $color(blue)(a^m/a^ n=a^(m-n)$

Applying the above to the exponents of $x$ $x$ and $y$ $y$

$= (\frac{81}{12}) . x^{28 - 1} . y^{12 - 3}$ $= (81/12) . x^(28-1) . y^(12-3$
$= \frac{27}{4} x^{27} . y^{9}$ $= color(blue)(27/4x^27.y^9$

How do you simplify (3x7y3)412xy3(3x^7y^3)^4/(12xy^3)?