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 .
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.
True, because the Negative Exponent Rule requires that the number be rewritten as a negative value.

Answer 1

To evaluate the equation \( 10^{-13} = -10^{-\sqrt{3}} \) and determine its truthfulness, we can analyze each side using properties of exponents.

The left side, \( 10^{-13} \), can be rewritten using the property of negative exponents: \[ 10^{-13} = \frac{1}{10^{13}} \]

For the right side, \( -10^{-\sqrt{3}} \): \[ -10^{-\sqrt{3}} = -\frac{1}{10^{\sqrt{3}}} \]

Now we can rewrite the original equation: \[ \frac{1}{10^{13}} \neq -\frac{1}{10^{\sqrt{3}}} \] Since the left side is a positive value and the right side is negative, these two expressions cannot be equal.

Thus, the equation \( 10^{-13} = -10^{-\sqrt{3}} \) is false.

So the correct response is: False, because according to the Negative Exponent Rule requires that the number be rewritten as a quotient, not as a negative value.