Question

use the properties of exponents to determine whether the equation 10^(-(1)/(3)) = -\root(3)(10) is true or false, select the response that correctly answers the question and provides the appropriate justification.

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^(-(1)/(3)) = -(1)/(\root(3)(10))
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 writting an equivalent expression

Answer 1

just choose a, b, c, or d

Answer 2

To determine whether the equation \( 10^{-\frac{1}{3}} = -\sqrt[3]{10} \) is true or false, we need to analyze each side of the equation.

The left side, \( 10^{-\frac{1}{3}} \), means \( \frac{1}{10^{\frac{1}{3}}} \) according to the negative exponent rule, which states that \( a^{-n} = \frac{1}{a^{n}} \).

The right side, \( -\sqrt[3]{10} \), is simply the negative value of the cube root of 10.

Now let's compare both sides:
- The left side \( 10^{-\frac{1}{3}} = \frac{1}{\sqrt[3]{10}} \).
- The right side is \( -\sqrt[3]{10} \).

Clearly, \( \frac{1}{\sqrt[3]{10}} \) is positive, whereas \( -\sqrt[3]{10} \) is negative.

Thus, the two sides are not equal.

### Conclusion
The correct response is:

**False, because according to the negative exponent rule, the equation should be \( 10^{-\frac{1}{3}} = \frac{1}{\sqrt[3]{10}} \), which is not equal to \( -\sqrt[3]{10} \).**

Answer 3

The correct answer is **B**: false because according to the negative exponent rule, the equation should be \( 10^{-\frac{1}{3}} = -\frac{1}{\sqrt[3]{10}} \).