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 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.

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.

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 sign moves to the front when writing an equivalent expression.

GPT-4o mini · Answer

To determine whether the equation $ 10^{-13} = -10^{-\sqrt{3}} $ is true or false, we should analyze each side of the equation carefully using properties of exponents. 1. The left side of the equation is $ 10^{-13} $. According to the Negative Exponent Rule, this can be rewritten as: $$ 10^{-13} = \frac{1}{10^{13}} $$ 2. The right side of the equation is $ -10^{-\sqrt{3}} $. Again, using the Negative Exponent Rule: $$ -10^{-\sqrt{3}} = -\frac{1}{10^{\sqrt{3}}} $$ To assess whether these two sides are equal, we can analyze the behavior of the terms: - The left side, $ \frac{1}{10^{13}} $, is a positive number (since $ 10^{13} $ is a very large positive number). - The right side, $ -\frac{1}{10^{\sqrt{3}}} $, is a negative number (since it's in the form of a negative fraction). Since one side is positive and the other side is negative, these two expressions cannot be equal. Thus, the statement $ 10^{-13} = -10^{-\sqrt{3}} $ is **false**. The best response is: **False, because according to the Negative Exponent Rule, the equation should be $ 10^{-13} = -\frac{1}{10^{\sqrt{3}}} $.**

I know the answer · Answer

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

GPT-4o mini · Answer

The statement you're suggesting can indeed be a way to express the reasoning behind why the equation $ 10^{-13} = -10^{-\sqrt{3}} $ is false.

Let’s break it down further:

Negative Exponent Rule: The Negative Exponent Rule states that a negative exponent can be rewritten as a reciprocal. For example: \[ a^{-n} = \frac{1}{a^n} \]
Application to the Problem:
- Left Side: \[ 10^{-13} = \frac{1}{10^{13}} \quad (\text{positive value}) \]
- Right Side: \[ -10^{-\sqrt{3}} = -\frac{1}{10^{\sqrt{3}}} \quad (\text{negative value}) \]
Comparison: Since the left side (a positive number) cannot equal the right side (a negative number), the equation is false.

So, the response "False, because the Negative Exponent Rule requires that the number be rewritten as a quotient, not as a negative value." is acceptable, but it primarily highlights the transformation rather than the conclusion about the values being positive and negative.

Given the context, I would suggest stating something like:

False, because according to the Negative Exponent Rule, $ 10^{-13} $ is a positive value while $ -10^{-\sqrt{3}} $ is a negative value, so they cannot be equal.

But, the original response you suggested captures an important aspect of working with negative exponents, even if it could clarify that the positivity or negativity of the result makes the equation false.