Let's simplify the algebraic expression [tex]\(\left(27 x^{\frac{5}{3}}\right)^{-\frac{1}{3}}\)[/tex].
1. Exponentiation Properties:
We start with the given expression
[tex]\[
\left(27 x^{\frac{5}{3}}\right)^{-\frac{1}{3}}
\][/tex]
According to the exponentiation properties, we can apply the exponent [tex]\(-\frac{1}{3}\)[/tex] to both the constant [tex]\(27\)[/tex] and the [tex]\(x^{\frac{5}{3}}\)[/tex] term separately.
2. Simplifying the Constant Term:
The term [tex]\(27\)[/tex] can be rewritten as [tex]\(3^3\)[/tex]:
[tex]\[
27 = 3^3
\][/tex]
Now apply the exponent:
[tex]\[
(3^3)^{-\frac{1}{3}} = 3^{3 \cdot -\frac{1}{3}} = 3^{-1} = \frac{1}{3}
\][/tex]
3. Simplifying the Variable Term:
Now let's handle [tex]\(x^{\frac{5}{3}}\)[/tex] raised to the power of [tex]\(-\frac{1}{3}\)[/tex]:
[tex]\[
\left(x^{\frac{5}{3}}\right)^{-\frac{1}{3}} = x^{\frac{5}{3} \cdot -\frac{1}{3}} = x^{-\frac{5}{9}}
\][/tex]
4. Combining the Results:
Combining the simplified constant and variable parts gives us:
[tex]\[
\frac{1}{3} \cdot x^{-\frac{5}{9}} = \frac{1}{3 x^{\frac{5}{9}}}
\][/tex]
Thus, the algebraic expression [tex]\(\left(27 x^{\frac{5}{3}}\right)^{-\frac{1}{3}}\)[/tex] simplifies to:
[tex]\[
\boxed{\frac{1}{3 x^{\frac{5}{9}}}}
\][/tex]
This matches the first choice in the given list.
