Question

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

Answer 1

To determine whether the equation \( 10^{-1/3} = -10^{-\sqrt{3}} \) is true or false, we can simplify both sides of the equation.

1. **Left Side**:
\[
10^{-1/3} = \frac{1}{10^{1/3}} = \frac{1}{\sqrt[3]{10}}
\]

2. **Right Side**:
\[
-10^{-\sqrt{3}} = -\frac{1}{10^{\sqrt{3}}} = -\frac{1}{10^{\sqrt{3}}}
\]

Now, let’s compare the two results:

- The left side, \( 10^{-1/3} \), is positive because it is the reciprocal of a positive number.
- The right side, \( -10^{-\sqrt{3}} \), is negative because it is the negative of a positive number.

Since one side is positive and the other side is negative, we can conclude that:

**The equation \( 10^{-1/3} = -10^{-\sqrt{3}} \) is false.**