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.(1 point)

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.(1 point)
Responses

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 superscript negative Start Fraction 1 over 3 End Fraction baseline equals negative Start Fraction 1 over root index 3 Start Root 10 End Root End Fraction .

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.

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.

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.

GPT-4o mini · Answer

Let's analyze the given equation $10^{-13} = -10^{-\sqrt{3}}$. First, let's rewrite each side of the equation using the properties of exponents: 1. According to the Negative Exponent Rule: $$ 10^{-13} = \frac{1}{10^{13}} $$ 2. For the right-hand side, we need to consider the negative sign separately: $$ -10^{-\sqrt{3}} = -\frac{1}{10^{\sqrt{3}}} $$ Now, let's compare both sides: - The left-hand side is $\frac{1}{10^{13}}$. - The right-hand side is $-\frac{1}{10^{\sqrt{3}}}$. Since $ \frac{1}{10^{13}} $ is a positive number and $ -\frac{1}{10^{\sqrt{3}}} $ is clearly negative (as it has a negative sign), these two sides cannot be equal. Thus, the equation $ 10^{-13} = -10^{-\sqrt{3}} $ is **False**. **Final Answer:** False, because the Negative Exponent Rule requires that the number be rewritten as a quotient, not as a negative value.