Let's simplify the expression [tex]\(\left(x^{-3}\right)^{\frac{1}{3}}\)[/tex] step-by-step.
1. Understand the expression: We have [tex]\((x^{-3})^{\frac{1}{3}}\)[/tex].
2. Apply the property of exponents: When you have an expression in the form [tex]\((a^m)^n\)[/tex], it simplifies to [tex]\(a^{m \cdot n}\)[/tex].
3. Calculate the exponent:
- Here, [tex]\(a = x\)[/tex], [tex]\(m = -3\)[/tex], and [tex]\(n = \frac{1}{3}\)[/tex].
- The new exponent will be [tex]\(m \cdot n = -3 \cdot \frac{1}{3} = -1\)[/tex].
4. Rewrite the expression:
[tex]\[
(x^{-3})^{\frac{1}{3}} = x^{-1}
\][/tex]
5. Recognize the equivalent forms:
- We know that [tex]\(x^{-1} = \frac{1}{x}\)[/tex].
Thus, the simplest form of the expression [tex]\(\left(x^{-3}\right)^{\frac{1}{3}}\)[/tex] is [tex]\(x^{-1}\)[/tex], which can also be written as [tex]\(\frac{1}{x}\)[/tex].
Correct answers:
- [tex]\(x^{-1}\)[/tex]
- [tex]\(\frac{1}{x}\)[/tex]
The other options [tex]\(x^9\)[/tex] and [tex]\(\frac{1}{x^9}\)[/tex] do not match the simplified form of the given expression.
