# How do you simplify ( 9 / 49) ^ (- 3 / 2)?

Jan 18, 2016

=27/(343

#### Explanation:

As per property:
(a/b)^color(blue)(m)= a^color(blue)(m)/(b^color(blue)(m

Applying the above to the expression :

(9/49)^ (-3/2) = 9^color(blue)(-3/2)/(49^color(blue)(-3/2

(3^2)^(color(blue)(-3/2))/((7^2)^color(blue)(-3/2

=(3^cancel2)^(-3/cancel2)/((7^cancel2)^(-3/cancel2)

$\textcolor{b l u e}{\text{~~~~~~~~~~~~~~Tony B Formatting test~~~~~~~~~~~~~~~~~~~}}$
$\left({3}^{\cancel{2}}\right) \left(\frac{3}{\cancel{2}}\right)$

${\left({3}^{\cancel{2}}\right)}^{\frac{3}{\cancel{2}}}$

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$\textcolor{b l u e}{\text{'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}}$

=3^-3/(7^-3

$= \frac{\frac{1}{27}}{\frac{1}{343}}$

$= \frac{343}{27}$

Jan 18, 2016

${\left(\frac{9}{49}\right)}^{- \frac{3}{2}} = {\left[{\left(\frac{3}{7}\right)}^{2}\right]}^{- \frac{3}{2}} = {\left(\frac{3}{7}\right)}^{-} 3 = {\left(\frac{7}{3}\right)}^{3} = \frac{343}{27}$

#### Explanation:

The minus in front of the index is instruction that this is a reciprocal

So we have: $\frac{1}{{\left(\frac{9}{49}\right)}^{\frac{3}{2}}}$

This is $\frac{{\left(49\right)}^{\frac{3}{2}}}{{\left(9\right)}^{\frac{3}{2}}}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Consider $\textcolor{w h i t e}{. .} {9}^{\frac{3}{2}}$

This is the same as ${\left(\sqrt{9} \textcolor{w h i t e}{.}\right)}^{3} = {3}^{3} = 27$

Giving: $\frac{{\left(49\right)}^{\frac{3}{2}}}{27}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Consider: ${49}^{\frac{3}{2}}$

This is the same as ${\left(\sqrt{49}\right)}^{3} = {7}^{3} = 343$

Giving:$\frac{343}{27} = 12 \frac{19}{27}$