Let's simplify the expression [tex]\(\left(x^3\right)^{-\frac{2}{3}}\)[/tex] step by step.
1. Understanding the Expression:
The given expression is [tex]\(\left(x^3\right)^{-\frac{2}{3}}\)[/tex]. Here, you have an exponentiation operation where [tex]\(x^3\)[/tex] is raised to the power of [tex]\(-\frac{2}{3}\)[/tex].
2. Properties of Exponents:
According to the properties of exponents, [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. Applying this property to our expression:
[tex]\[
\left(x^3\right)^{-\frac{2}{3}} = x^{3 \cdot \left(-\frac{2}{3}\right)}
\][/tex]
3. Simplifying Multiplication of Exponents:
Now, we need to multiply the exponents [tex]\(3\)[/tex] and [tex]\(-\frac{2}{3}\)[/tex]:
[tex]\[
3 \cdot \left(-\frac{2}{3}\right) = -2
\][/tex]
4. Final Expression:
Substituting [tex]\(-2\)[/tex] back into the expression, we get:
[tex]\[
x^{-2}
\][/tex]
Thus, the simplified form of [tex]\(\left(x^3\right)^{-\frac{2}{3}}\)[/tex] is [tex]\(x^{-2}\)[/tex]. However, in the final numerical approach, as given, it is realized simply as [tex]\((x^3)^{-0.666666666666667}\)[/tex]. Both are equivalent since [tex]\(-\frac{2}{3}\)[/tex] is numerically equal to [tex]\(-0.666666666666667\)[/tex].
