Use the properties of exponents to determine whether the equation 10−13=−10−−√3

Question

is true or false. Select the response that correctly answers the question and provides the appropriate justification. (1 point)
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.

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 raised to the negative 1 third power is equal to negative the fraction with numerator 1 and denominator cube root of 10

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

Answer 1

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

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

2. Right Side: \( -10^{-\frac{1}{3}} \) can also be simplified using the Negative Exponent Rule: \[ -10^{-\frac{1}{3}} = -\frac{1}{10^{\frac{1}{3}}} \]

Next, we look at the values of each term:

• \( 10^{-13} \) gives a very small positive number (specifically, it's \( \frac{1}{10^{13}} \)).
• \( -10^{-\frac{1}{3}} \) gives a negative value because it is a negative sign multiplied by a fraction.

Since \( 10^{-13} \) is positive and \( -10^{-\frac{1}{3}} \) is negative, we can conclude that: \[ 10^{-13} \neq -10^{-\frac{1}{3}} \]

Thus, the equation is false.

The correct response is:

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