Let's simplify the given expression step-by-step:
[tex]\[
\left(\frac{4^{\frac{5}{4}} \cdot 4^{\frac{1}{4}}}{4^{\frac{1}{2}}}\right)^{\frac{1}{2}}
\][/tex]
First, we simplify the expression inside the parentheses. We start with the numerator:
[tex]\[
4^{\frac{5}{4}} \cdot 4^{\frac{1}{4}}
\][/tex]
When multiplying powers with the same base, we add the exponents:
[tex]\[
4^{\frac{5}{4}} \cdot 4^{\frac{1}{4}} = 4^{\frac{5}{4} + \frac{1}{4}} = 4^{\frac{6}{4}} = 4^{\frac{3}{2}}
\][/tex]
Now we substitute [tex]\( 4^{\frac{3}{2}} \)[/tex] into the original expression:
[tex]\[
\left(\frac{4^{\frac{3}{2}}}{4^{\frac{1}{2}}}\right)^{\frac{1}{2}}
\][/tex]
Next, we simplify the division inside the parentheses. When dividing powers with the same base, we subtract the exponents:
[tex]\[
\frac{4^{\frac{3}{2}}}{4^{\frac{1}{2}}} = 4^{\frac{3}{2} - \frac{1}{2}} = 4^{\frac{2}{2}} = 4^1 = 4
\][/tex]
Now we have:
[tex]\[
(4)^{\frac{1}{2}}
\][/tex]
Taking the square root (equivalent to raising to the power of [tex]\(\frac{1}{2}\)[/tex]):
[tex]\[
4^{\frac{1}{2}} = \sqrt{4} = 2
\][/tex]
Therefore, the expression equivalent to [tex]\(\left(\frac{4^{\frac{5}{4}} \cdot 4^{\frac{1}{4}}}{4^{\frac{1}{2}}}\right)^{\frac{1}{2}}\)[/tex] is:
[tex]\[
2
\][/tex]
So, the correct answer is 2.
