Use the properties of exponents to determine whether the equation 10−1/3=−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 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 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 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 10−1/ 3=−1/10√3.

Answer 1

To determine whether the equation \( 10^{-1/3} = -10^{-\sqrt{3}} \) is true or false, let's analyze both sides separately using the properties of exponents.

1. Left Side: \( 10^{-1/3} \) Using the Negative Exponent Rule, which states that \( a^{-n} = \frac{1}{a^n} \), we can rewrite this expression as: \[ 10^{-1/3} = \frac{1}{10^{1/3}}. \]

2. Right Side: \( -10^{-\sqrt{3}} \) Similarly, using the Negative Exponent Rule again, we have: \[ -10^{-\sqrt{3}} = -\frac{1}{10^{\sqrt{3}}}. \]

Now, we need to compare these two expressions:

• The left side simplifies to \( \frac{1}{10^{1/3}} \).
• The right side simplifies to \( -\frac{1}{10^{\sqrt{3}}} \).

Since \( \frac{1}{10^{1/3}} \) is a positive quantity and \( -\frac{1}{10^{\sqrt{3}}} \) is a negative quantity, we can conclude that \( 10^{-1/3} \) is not equal to \( -10^{-\sqrt{3}} \).

Therefore, the correct response is:

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