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

Question

10
−
1
3
=
−
10
3
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.

False, because according to the Negative Exponent Rule, the equation should be 10−13=−110√3
10
−
1
3
=
−
1
10

Answer 1

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

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

2. Right Side: \[ -10^{-\frac{1}{3}} = -\frac{1}{10^{\frac{1}{3}}} \] This is the same as: \[ -\frac{1}{\sqrt[3]{10}} \]

Now we need to compare both sides:

• The left side \(10^{-13} = \frac{1}{10^{13}}\) is a positive number (since \(10^{13}\) is a positive number, and its reciprocal is also positive).
• The right side \(-10^{-\frac{1}{3}} = -\frac{1}{\sqrt[3]{10}}\) is a negative number (since it has a negative sign).

Since one side is positive and the other side is negative, the two sides are not equal.

Thus, the equation is False.

Justification: The left side is positive while the right side is negative, hence they cannot be equal.

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