Find x, if : ( 3 5 )^ x+1 =125 27 - Vidyadip

Question

Find $x$, if : $\left(\sqrt{\frac{3}{5}}\right)^{x+1}=\frac{125}{27}$

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Answer

$\left(\sqrt{\frac{3}{5}}\right)^{x+1}=\frac{125}{27}$
$ \Rightarrow\left[\left(\frac{3}{5}\right)^{\frac{1}{2}}\right]^{x+1}=\frac{5 \times 5 \times 5}{3 \times 3 \times 3}$
$\Rightarrow\left(\frac{3}{5}\right)^{\frac{x+1}{2}}=\left(\frac{5}{3}\right)^3$
$\Rightarrow\left(\frac{3}{5}\right)^{\frac{x+1}{2}}=\left(\frac{3}{5}\right)^{-} 3$
We know that if bases are equal, the powers are equal
$\Rightarrow \frac{x+1}{2}=-3$
$\Rightarrow x+1=-6$
$ \Rightarrow x=-6-1$
$ \Rightarrow x=-7$

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