Question

Use the properties of exponents to determine whether the equation 10−13=−10−−√3 is true or false. Select the response that correctly answers the question and provides the appropriate justification.

Answer 1

To evaluate the equation \( 10^{-13} = -10^{-\sqrt{3}} \), we need to analyze both sides of the equation using the properties of exponents.

Starting with the left side:

\[
10^{-13} = \frac{1}{10^{13}}
\]

Now, let’s evaluate the right side.

The term \( -10^{-\sqrt{3}} \) can be rewritten as:

\[
-10^{-\sqrt{3}} = -\frac{1}{10^{\sqrt{3}}}
\]

Since \( \sqrt{3} \) is a positive number, \( 10^{\sqrt{3}} \) is also positive. Hence, \( -\frac{1}{10^{\sqrt{3}}} \) is a negative number.

Now we compare both sides:

- The left side \( 10^{-13} = \frac{1}{10^{13}} \) is a positive number.
- The right side \( -10^{-\sqrt{3}} = -\frac{1}{10^{\sqrt{3}}} \) is a negative number.

Since a positive number cannot equal a negative number, the equation \( 10^{-13} = -10^{-\sqrt{3}} \) is false.

Thus, the final answer is **False**.