Solve the equation $(\frac{1}{5})^{-2}\times125^{x}=\sqrt{25^{x}}$ . Answer x = ...

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Worked solution

1. Express every term using base $5$ and simplify the square root.

$(\frac{1}{5})^{-2}\times125^x=\sqrt{25^x}$
$5^2\times(5^3)^x=\sqrt{(5^2)^x}$
$5^2\times5^{3x}=\sqrt{5^{2x}}$
$5^{2+3x}=5^x$

2. Equate indices and solve for $x$.

Since the bases are equal,
$2+3x=x$
$2=-2x$
$x=-1$

Answer

$x=1$

AI analysis — may contain mistakes.

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