To determine whether the equation \( 10^{(-1/3)} = -\sqrt[3]{10} \) is true or false, we can apply the properties of exponents and roots.
First, we rewrite \( 10^{(-1/3)} \) using the Negative Exponent Rule:
\[ 10^{(-1/3)} = \frac{1}{10^{1/3}} \]
Next, recognize that \( 10^{1/3} \) can be rewritten as the cube root of \( 10 \):
\[ 10^{1/3} = \sqrt[3]{10} \]
So, we have:
\[ 10^{(-1/3)} = \frac{1}{\sqrt[3]{10}} \]
Now, we need to compare \( \frac{1}{\sqrt[3]{10}} \) and \(-\sqrt[3]{10}\):
- \( \frac{1}{\sqrt[3]{10}} \) is a positive quantity, since the cube root of a positive number is positive.
- \(-\sqrt[3]{10}\) is negative because it has a negative sign in front.
Since the left side is positive and the right side is negative, the equation \( 10^{(-1/3)} = -\sqrt[3]{10} \) is false.
Thus, the correct response is:
False, because according to the Negative Exponent Rule, the equation should be \( 10^{(-1/3)} = \frac{1}{\sqrt[3]{10}} \), and this cannot equal \(-\sqrt[3]{10}\).