#### Answer

$x=\dfrac{4}{3}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given equation, $
\left( \sqrt{2}\right)^{x+4}=4^x
,$ express both sides of the equation in the same base. Once the bases are the same, equate the exponents. Use the properties of equality to isolate the variable.
$\bf{\text{Solution Details:}}$
Using the definition of rational exponents which is given by $a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\left( 2^{\frac{1}{2}}\right)^{x+4}=4^x
.\end{array}
Since $4=2^2$, the equation above is equivalent to
\begin{array}{l}\require{cancel}
\left( 2^{\frac{1}{2}}\right)^{x+4}=(2^2)^x
.\end{array}
Using the Power Rule of the laws of exponents which is given by $\left( x^m \right)^p=x^{mp},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
2^{\frac{1}{2}(x+4)}=2^{2(x)}
\\\\
2^{\frac{1}{2}x+2}=2^{2x}
.\end{array}
Since the bases are the same, then the exponents can be equated. Hence,
\begin{array}{l}\require{cancel}
\frac{1}{2}x+2=2x
.\end{array}
Using the properties of equality to isolate the variable, the equation above is equivalent to
\begin{array}{l}\require{cancel}
2\left( \frac{1}{2}x+2 \right)=(2x)2
\\\\
x+4=4x
\\\\
x-4x=-4
\\\\
-3x=-4
\\\\
x=\dfrac{-4}{-3}
\\\\
x=\dfrac{4}{3}
.\end{array}