Answer:
[tex]2^{\frac{5}{12}}[/tex]
Step-by-step explanation:
- The original expression [tex]\sqrt{2^5} ^{\frac{1}{4}}[/tex] can be transformed into [tex](2^{\frac{5}{3}})^{\frac{1}{4}}[/tex] : both expressions are equivalent, the root of certain number is equivalent to that number power at a fraction whose denominator is the index of the root. The simpliest example for this statement is [tex]\sqrt{x} =x^{\frac{1}{2}}[/tex] (the squared root of x equals x raised at 1/2).
- Now, the expression[tex](2^{\frac{5}{3}})^{\frac{1}{4}}[/tex] can be simplified by using the power of a power property, which simply states that if [tex]b\neq 0[/tex] and [tex]((b)^n)^m=b^{n\times{m}}[/tex]. In this case, then [tex](2^{\frac{5}{3}})^{\frac{1}{4}}=2^{\frac{5}{3}\times{\frac{1}{4}}}=2^{\frac{5}{12}}[/tex], which is the final expression.
