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

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 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.

Answer 1

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

Starting with the left side: \[ 10^{-13} = \frac{1}{10^{13}} \]

Now for the right side: \[ -10^{-\frac{1}{3}} = -\frac{1}{10^{\frac{1}{3}}} \]

Now, we need to determine the two sides:

1. The left side is \( \frac{1}{10^{13}} \).
2. The right side, after simplification, is \( -\frac{1}{10^{\frac{1}{3}}} \).

Clearly, \( \frac{1}{10^{13}} \) is a positive value, while \( -\frac{1}{10^{\frac{1}{3}}} \) is negative. Hence, the left side is not equal to the right side.

Therefore, the statement \( 10^{-13} = -10^{-\frac{1}{3}} \) is false.

The correct response is:

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