Use the properties of exponents to determine whether the equation 10^-1/3=-3√10 is true or false. Select the response that correctly answers the question and provides the appropriate justification.(1 point)

To determine whether the equation \( 10^{-1/3} = -3\sqrt{10} \) is true or false, let's analyze the left side using the properties of exponents.

1. The negative exponent rule states that \( a^{-n} = \frac{1}{a^n} \). Therefore:
\[
10^{-1/3} = \frac{1}{10^{1/3}} = \frac{1}{\sqrt[3]{10}}
\]

2. Now, let’s compare \( \frac{1}{\sqrt[3]{10}} \) to \( -3\sqrt{10} \).

To clarify:
- \( -3\sqrt{10} \) is a negative value.
- \( \frac{1}{\sqrt[3]{10}} \) is a positive value, since it represents the reciprocal of a cube root, which cannot be negative.

Therefore, \( 10^{-1/3} \) cannot be equal to \( -3\sqrt{10} \) since one side is positive and the other is negative.

As a result, the correct response is:

A) False, because the Negative Exponent Rule requires that the number be rewritten as a quotient, not as a negative value.