To determine which expression is equal to 16, we can evaluate each option one by one.
A. \(\left(6^{\frac{1}{3}}\right)^{3}\)
Using the exponent rule \( (a^{m})^{n} = a^{m \cdot n} \): \[ (6^{\frac{1}{3}})^{3} = 6^{\frac{1}{3} \cdot 3} = 6^{1} = 6 \]
B. \(\left(8^{\frac{1}{2}}\right)^{2}\)
\[ (8^{\frac{1}{2}})^{2} = 8^{\frac{1}{2} \cdot 2} = 8^{1} = 8 \]
C. \(\left(8^{\frac{1}{4}}\right)^{4}\)
\[ (8^{\frac{1}{4}})^{4} = 8^{\frac{1}{4} \cdot 4} = 8^{1} = 8 \]
D. \(\left(16^{\frac{1}{3}}\right)^{3}\)
\[ (16^{\frac{1}{3}})^{3} = 16^{\frac{1}{3} \cdot 3} = 16^{1} = 16 \]
E. \(\left(64^{\frac{1}{4}}\right)^{4}\)
\[ (64^{\frac{1}{4}})^{4} = 64^{\frac{1}{4} \cdot 4} = 64^{1} = 64 \]
After evaluating all the expressions, we can see that the option that equals 16 is:
D. \(\left(16^{\frac{1}{3}}\right)^{3}\)