Answer :
To find which expression is equivalent to \(\left(\frac{4^{\frac{5}{4}} \cdot 4^{\frac{1}{4}}}{4^{\frac{1}{2}}}\right)^{\frac{1}{2}}\), let's break it down step-by-step.
1. Combine the exponents in the numerator:
[tex]\[ 4^{\frac{5}{4}} \cdot 4^{\frac{1}{4}} = 4^{\left(\frac{5}{4} + \frac{1}{4}\right)} = 4^{\frac{6}{4}} = 4^{\frac{3}{2}} \][/tex]
So our expression now is:
[tex]\[ \left(\frac{4^{\frac{3}{2}}}{4^{\frac{1}{2}}}\right)^{\frac{1}{2}} \][/tex]
2. Simplify the division of the bases with exponents:
[tex]\[ \frac{4^{\frac{3}{2}}}{4^{\frac{1}{2}}} = 4^{\left(\frac{3}{2} - \frac{1}{2}\right)} = 4^{\frac{2}{2}} = 4^1 = 4 \][/tex]
Now our expression simplifies to:
[tex]\[ \left(4\right)^{\frac{1}{2}} \][/tex]
3. Take the square root of the result:
[tex]\[ \left(4\right)^{\frac{1}{2}} = \sqrt{4} = 2 \][/tex]
Thus, the equivalent expression is:
[tex]\[ 2 \][/tex]
So the correct answer is:
[tex]\[ \boxed{2} \][/tex]
1. Combine the exponents in the numerator:
[tex]\[ 4^{\frac{5}{4}} \cdot 4^{\frac{1}{4}} = 4^{\left(\frac{5}{4} + \frac{1}{4}\right)} = 4^{\frac{6}{4}} = 4^{\frac{3}{2}} \][/tex]
So our expression now is:
[tex]\[ \left(\frac{4^{\frac{3}{2}}}{4^{\frac{1}{2}}}\right)^{\frac{1}{2}} \][/tex]
2. Simplify the division of the bases with exponents:
[tex]\[ \frac{4^{\frac{3}{2}}}{4^{\frac{1}{2}}} = 4^{\left(\frac{3}{2} - \frac{1}{2}\right)} = 4^{\frac{2}{2}} = 4^1 = 4 \][/tex]
Now our expression simplifies to:
[tex]\[ \left(4\right)^{\frac{1}{2}} \][/tex]
3. Take the square root of the result:
[tex]\[ \left(4\right)^{\frac{1}{2}} = \sqrt{4} = 2 \][/tex]
Thus, the equivalent expression is:
[tex]\[ 2 \][/tex]
So the correct answer is:
[tex]\[ \boxed{2} \][/tex]
