To determine whether the equation \( 10^{-1/3} = -\sqrt[3]{10} \) is true or false, we can simplify and analyze both sides of the equation.
**Step 1: Simplify the left side.**
The left side is \( 10^{-1/3} \). According to the Negative Exponent Rule:
\[
a^{-b} = \frac{1}{a^b}
\]
So, we can rewrite \( 10^{-1/3} \) as:
\[
10^{-1/3} = \frac{1}{10^{1/3}} = \frac{1}{\sqrt[3]{10}}
\]
**Step 2: Evaluate the right side.**
The right side of the equation is \( -\sqrt[3]{10} \).
**Step 3: Compare the two sides.**
Now we have:
\[
10^{-1/3} = \frac{1}{\sqrt[3]{10}} \quad \text{(left side)}
\]
\[
-\sqrt[3]{10} \quad \text{(right side)}
\]
**Step 4: Analyze the values.**
We can see:
- The left side \( \frac{1}{\sqrt[3]{10}} \) is a positive number because \(\sqrt[3]{10}\) is a positive number (any real cube root is positive).
- The right side \( -\sqrt[3]{10} \) is a negative number, as it has a negative sign in front.
Since the left side is positive and the right side is negative, the two sides cannot be equal.
**Conclusion:**
The equation \( 10^{-1/3} = -\sqrt[3]{10} \) is **false**. This is because the Negative Exponent Rule allows us to rewrite \( 10^{-1/3} \) as \( \frac{1}{\sqrt[3]{10}} \), which is positive and cannot equal the negative value \( -\sqrt[3]{10} \).
**Selected response:**
False, because the Negative Exponent Rule requires that the number be rewritten as a quotient, not as a negative value.