To determine which expression is equal to [tex]\( 8 \)[/tex], let's evaluate each option step-by-step:
Option A:
[tex]\[ \left(2^{\frac{1}{4}}\right)^4 \][/tex]
First, we calculate the inner expression:
[tex]\[ 2^{\frac{1}{4}} \][/tex]
Then raise it to the fourth power:
[tex]\[ \left(2^{\frac{1}{4}}\right)^4 = (2^{\frac{1}{4} \cdot 4}) = 2^1 = 2 \][/tex]
This gives us a result of 2, which is not equal to 8.
Option B:
[tex]\[ \left(4^{\frac{1}{2}}\right)^2 \][/tex]
First, we calculate the inner expression:
[tex]\[ 4^{\frac{1}{2}} = \sqrt{4} = 2 \][/tex]
Then raise it to the second power:
[tex]\[ \left(4^{\frac{1}{2}}\right)^2 = 2^2 = 4 \][/tex]
This gives us a result of 4, which is not equal to 8.
Option C:
[tex]\[ \left(6^{\frac{1}{3}}\right)^3 \][/tex]
First, we calculate the inner expression:
[tex]\[ 6^{\frac{1}{3}} \][/tex]
Then raise it to the third power:
[tex]\[ \left(6^{\frac{1}{3}}\right)^3 = (6^{\frac{1}{3} \cdot 3}) = 6^1 = 6 \][/tex]
This gives us a result of 6, which is not equal to 8.
Option D:
[tex]\[ \left(8^{\frac{1}{2}}\right)^2 \][/tex]
First, we calculate the inner expression:
[tex]\[ 8^{\frac{1}{2}} = \sqrt{8} \][/tex]
Then raise it to the second power:
[tex]\[ \left(8^{\frac{1}{2}}\right)^2 = (\sqrt{8})^2 = 8 \][/tex]
This gives us a result of 8, which is indeed equal to 8. Therefore, this option satisfies the condition.
Option E:
[tex]\[ \left(16^{\frac{1}{4}}\right)^4 \][/tex]
First, we calculate the inner expression:
[tex]\[ 16^{\frac{1}{4}} \][/tex]
Then raise it to the fourth power:
[tex]\[ \left(16^{\frac{1}{4}}\right)^4 = (16^{\frac{1}{4} \cdot 4}) = 16^1 = 16 \][/tex]
This gives us a result of 16, which is not equal to 8.
Thus, the expression that is equal to 8 is:
[tex]\[ \boxed{D} \][/tex]
