To find the value of the expression [tex]\(\left(8^{\frac{5}{3}}\right)^{\frac{1}{5}}\)[/tex], we need to simplify it step by step.
Starting with the inner part of the expression:
[tex]\[ 8^{\frac{5}{3}} \][/tex]
First, recall that 8 can be written as a power of 2:
[tex]\[ 8 = 2^3 \][/tex]
So we can rewrite the expression as:
[tex]\[ 8^{\frac{5}{3}} = (2^3)^{\frac{5}{3}} \][/tex]
Using the property of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we can simplify it to:
[tex]\[ (2^3)^{\frac{5}{3}} = 2^{3 \cdot \frac{5}{3}} = 2^5 \][/tex]
Now, we'd need to take this result to the power of [tex]\(\frac{1}{5}\)[/tex]:
[tex]\[ \left(2^5\right)^{\frac{1}{5}} \][/tex]
Using the same property of exponents, [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we have:
[tex]\[ (2^5)^{\frac{1}{5}} = 2^{5 \cdot \frac{1}{5}} = 2^1 = 2 \][/tex]
Hence, the value of the expression [tex]\(\left(8^{\frac{5}{3}}\right)^{\frac{1}{5}}\)[/tex] is:
[tex]\[
\boxed{2}
\][/tex]
This matches choice A.
