Certainly! Let's solve the 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]
### Step 1: Simplify the inner part of the expression
First, we need to simplify the base expression inside the parentheses: [tex]\[\frac{4^{\frac{5}{4}} \cdot 4^{\frac{1}{4}}}{4^{\frac{1}{2}}}\][/tex]
We start by simplifying the numerator:
[tex]\[4^{\frac{5}{4}} \cdot 4^{\frac{1}{4}}\][/tex]
Using the properties of exponents (same base), we can combine the exponents by adding them:
[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]
Next, we incorporate the denominator:
[tex]\[\frac{4^{\frac{3}{2}}}{4^{\frac{1}{2}}\][/tex]
Again, using exponent rules, we subtract the exponents when dividing:
[tex]\[4^{\frac{3}{2} - \frac{1}{2}} = 4^{1} = 4\][/tex]
### Step 2: Apply the outer exponent
Now we need to raise 4 to the power of [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[(4)^{\frac{1}{2}}\][/tex]
This is the same as taking the square root of 4:
[tex]\[\sqrt{4} = 2\][/tex]
### Step 3: Evaluate alternative expressions
Let's check the equivalency of our derived answer [tex]\(2\)[/tex] with the given options:
1. [tex]\(\sqrt[16]{4^5}\)[/tex]:
[tex]\[4^5 = 1024\][/tex]
[tex]\[\sqrt[16]{1024} = \sqrt[16]{2^{10}} = (2^{10})^{\frac{1}{16}} = 2^{\frac{10}{16}} = 2^{\frac{5}{8}} \approx 1.5422108254079407\][/tex]
Hence, this does not equal [tex]\(2\)[/tex].
2. [tex]\(\sqrt{2^5}\)[/tex]:
[tex]\[2^5 = 32\][/tex]
[tex]\[\sqrt{32} = 32^{\frac{1}{2}} = \sqrt{32} \approx 5.656854249492381\][/tex]
Hence, this does not equal [tex]\(2\)[/tex].
3. [tex]\(2\)[/tex]:
This is exactly what we derived from our original expression. Thus, this is correct.
4. [tex]\(4\)[/tex]:
Clearly, [tex]\(4\)[/tex] is not equal to our derived answer [tex]\(2\)[/tex].
Thus, the expression [tex]\(\left(\frac{4^{\frac{5}{4}} \cdot 4^{\frac{1}{4}}}{4^{\frac{1}{2}}}\right)^{\frac{1}{2}}\)[/tex] is equivalent to [tex]\(2\)[/tex].
