Sure, let’s solve the algebraic expression [tex]\(\left(27 x^{\frac{5}{3}}\right)^{-\frac{1}{3}}\)[/tex].
### Step-by-Step Solution:
1. Understand the given expression:
[tex]\[\left(27 x^{\frac{5}{3}}\right)^{-\frac{1}{3}}\][/tex]
2. Consider the expression inside the parentheses:
[tex]\[\left(27 x^{\frac{5}{3}}\right)\][/tex]
3. Apply the exponent rule [tex]\((a \cdot b)^m = a^m \cdot b^m\)[/tex]:
[tex]\[\left(27 x^{\frac{5}{3}}\right)^{-\frac{1}{3}} = 27^{-\frac{1}{3}} \cdot \left(x^{\frac{5}{3}}\right)^{-\frac{1}{3}}\][/tex]
4. Simplify the numerical part [tex]\(27^{-\frac{1}{3}}\)[/tex]:
- First recognize that [tex]\(27 = 3^3\)[/tex], then:
[tex]\[27^{-\frac{1}{3}} = (3^3)^{-\frac{1}{3}} = 3^{-1} = \frac{1}{3}\][/tex]
5. Simplify the exponent in [tex]\(x\)[/tex] using the power of a power property [tex]\((a^m)^n = a^{mn}\)[/tex]:
[tex]\[\left(x^{\frac{5}{3}}\right)^{-\frac{1}{3}} = x^{\left(\frac{5}{3} \cdot -\frac{1}{3}\right)} = x^{-\frac{5}{9}}\][/tex]
6. Combine the simplified parts:
[tex]\[\left(27 x^{\frac{5}{3}}\right)^{-\frac{1}{3}} = \frac{1}{3} \cdot x^{-\frac{5}{9}}\][/tex]
7. Final equivalent expression:
[tex]\[\frac{1}{3} x^{-\frac{5}{9}}\][/tex]
Thus, the equivalent expression for [tex]\(\left(27 x^{\frac{5}{3}}\right)^{-\frac{1}{3}}\)[/tex] is [tex]\(\frac{1}{3} x^{-\frac{5}{9}}\)[/tex].
