To determine whether the equation \(10^{-\frac{1}{3}} = -3^{\sqrt{10}}\) is true or false, let's analyze both sides of the equation.
1. **Left Side: \(10^{-\frac{1}{3}}\)**
By the Negative Exponent Rule, \(a^{-n} = \frac{1}{a^n}\). Therefore:
\[
10^{-\frac{1}{3}} = \frac{1}{10^{\frac{1}{3}}} = \frac{1}{\sqrt[3]{10}} \approx 0.464.
\]
2. **Right Side: \(-3^{\sqrt{10}}\)**
Here we deal with a negative sign in front, meaning:
\[
-3^{\sqrt{10}} \approx -3^{3.162} \text{ (since \(\sqrt{10} \approx 3.162\))}.
\]
The value of \(3^{3.162}\) is positive, thus:
\[
-3^{\sqrt{10}} < 0.
\]
Since the left side \(10^{-\frac{1}{3}}\) is approximately \(0.464\) (positive) and the right side \(-3^{\sqrt{10}}\) is negative, these two sides are not equal.
Thus, the equation is **False**.
The appropriate justification is:
**False, because the Negative Exponent Rule requires that the number be rewritten as a quotient, not as a negative value.**