Answer
$x=-\dfrac{1}{4}$
Work Step by Step
RECALL:
(i) If $a^M=a^N$, then $M=N.$
(ii) $\sqrt[n]{a}=a^{\frac{1}{n}}.$
(iii) $\dfrac{1}{a^m} = a^{-m}.$
Use rule (ii) above to obtain
$4^{x}=\dfrac{1}{2^{\frac{1}{2}}}.$
Use rule (iii) above to obtain
$4^x=2^{-\frac{1}{2}}.$
Write 4 as a power of 2 to obtain
$(2^2)^x = 2^{-\frac{1}{2}}
\\2^{2x} =2^{-\frac{1}{2}}.$
Use rule (i) above to obtain
$2x=-\frac{1}{2}.$
Solve the equation:
$\begin{array}{ccc}
&2x &= &-\frac{1}{2}
\\&\frac{1}{2} (2x) &= &-\dfrac{1}{2} \cdot \dfrac{1}{2}
\\&x &= &-\dfrac{1}{4}
\\\end{array}.$
