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

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 .

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.

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.
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Answer 1

To determine whether the equation \( 10^{-13} = -10^{-\frac{1}{3}} \) is true or false, let's analyze both sides of the equation using the properties of exponents.

1. The left side, \( 10^{-13} \), can be rewritten using the Negative Exponent Rule: \[ 10^{-13} = \frac{1}{10^{13}} \]

2. Now, for the right side \( -10^{-\frac{1}{3}} \): \[ -10^{-\frac{1}{3}} = -\frac{1}{10^{\frac{1}{3}}} \]

Now, we need to compare \( \frac{1}{10^{13}} \) and \( -\frac{1}{10^{\frac{1}{3}}} \).

Since \( 10^{13} \) is a very large positive number, \( \frac{1}{10^{13}} \) is a small positive number. On the other hand, \( -\frac{1}{10^{\frac{1}{3}}} \) is a negative number because of the negative sign.

Thus, \( \frac{1}{10^{13}} \) (which is positive) cannot equal \( -\frac{1}{10^{\frac{1}{3}}} \) (which is negative), meaning that the equation is false.

Therefore, the correct response is:

False, because according to the Negative Exponent Rule, the equation should be \( 10^{-13} = -\frac{1}{10^{\frac{1}{3}}} \).