If $$ x0 $$, which of the following expressions is equivalent to $$ \sqrt[3]{x^{-6}} $$ ?

Question

If $$ x>0 $$, which of the following expressions is equivalent to $$ \sqrt[3]{x^{-6}} $$ ?

UpStudy AI · Accepted Answer

To simplify the expression $$ \sqrt[3]{x^{-6}} $$ for $$ x > 0 $$, follow these steps:

1. **Rewrite the radical as an exponent:**
   $$
   \sqrt[3]{x^{-6}} = (x^{-6})^{1/3}
   $$

2. **Apply the power of a power rule $$(a^m)^n = a^{mn}$$:**
   $$
   (x^{-6})^{1/3} = x^{-6 \times \frac{1}{3}} = x^{-2}
   $$

3. **Express the negative exponent as a reciprocal:**
   $$
   x^{-2} = \frac{1}{x^2}
   $$

Therefore, an equivalent expression to $$ \sqrt[3]{x^{-6}} $$ is:

$$
\frac{1}{x^2}
$$

**Answer:** $$ \dfrac{1}{\,x^{2}\,} $$
An equivalent expression to $$ \sqrt[3]{x^{-6}} $$ is $$ \frac{1}{x^2} $$.

If \( x>0 \), which of the following expressions is equivalent to \( \sqrt[3]{x^{-6}} \) ?