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 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.

Answer 1

To determine whether the equation \( 10^{-13} = -10^{-\sqrt{3}} \) is true or false, we can analyze both sides of the equation using the properties of exponents.

1. Left Side: \[ 10^{-13} = \frac{1}{10^{13}} \]

2. Right Side: \[ -10^{-\sqrt{3}} = -\frac{1}{10^{\sqrt{3}}} \]

Now we can compare both sides:

• The left side, \( 10^{-13} \), evaluates to a positive number (specifically, a fraction).
• The right side, \( -10^{-\sqrt{3}} \), evaluates to a negative number because it contains a negative sign in front.

Since a positive number cannot equal a negative number, we conclude that the equation is false.

Now, assessing the justification:

• The correct response is: False, because according to the Negative Exponent Rule, the equation should be \( 10^{-13} = -\frac{1}{10^{\sqrt{3}}} \).

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