## College Algebra (11th Edition)

$x=\dfrac{4}{3}$
$\bf{\text{Solution Outline:}}$ 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}