Let's simplify one thing at a time to make some sense of that mess. Let's start with the cubed root of 8 to the first power. If we rewrite that using exponents we have: [tex]( \sqrt[3]{8^1} )=8 ^{ \frac{1}{3} } [/tex]. Now let's raise that to the power of (1/4)x: [tex](8 ^{ \frac{1}{3} } ) ^{ \frac{1}{4}x } [/tex]. The power rule of exponents says that we need to multiply those rational exponents together, and when we do, this is what we get: [tex]8 ^{ \frac{1}{12} x} [/tex]. The denominator of that rational exponent serves as the index for the radical, and the 1x serves as the power on the 8. So what we have after we put it back into radical form is [tex] \sqrt[12]{8^x} [/tex], third choice down.
