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. Responses True, because the Negative Exponent Rule requires that the number be rewritten as a negative value. True, because the Negative Exponent Rule requires that the number be rewritten as a negative value. - no response given True, because the negative sign moves to the front when writing an equivalent expression. True, because the negative sign moves to the front when writing an equivalent expression. - no response given False, because the Negative Exponent Rule requires that the number be rewritten as a quotient, not as a negative value. False, because the Negative Exponent Rule requires that the number be rewritten as a quotient, not as a negative value. - no response given False, because according to the Negative Exponent Rule, the equation should be 10−13=−110√3. False, because according to the Negative Exponent Rule, the equation should be 10 raised to the negative 1 third power is equal to negative the fraction with numerator 1 and denominator cube root of 10. - no response given

Answer 1

To determine whether the equation \( 10^{-13} = -10^{-\sqrt{3}} \) is true or false, let's simplify and analyze both sides of the equation using the properties of exponents.

1. **Left Side:**
\[
10^{-13} = \frac{1}{10^{13}}
\]

2. **Right Side:**
The term \( 10^{-\sqrt{3}} \) can also be rewritten using the Negative Exponent Rule:
\[
-10^{-\sqrt{3}} = -\frac{1}{10^{\sqrt{3}}}
\]

Now we can rewrite the entire equation:
\[
\frac{1}{10^{13}} = -\frac{1}{10^{\sqrt{3}}}
\]

Next, analyzing both sides:
- The left side \( \frac{1}{10^{13}} \) is a positive value (since the base 10 raised to any positive power is positive).
- The right side \( -\frac{1}{10^{\sqrt{3}}} \) is a negative value (since we have a negative sign in front of the fraction).

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

Therefore, the correct response is:
**False, because a positive number (left side) cannot equal a negative number (right side).**