Question

Use the properties of exponents to determine whether the equation 10 ^ (- 1/3) = - root(10, 3) is true or false. Select the response that correctly answers the question and provides the appropriate justification. (1 point) 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. False, because according to the Negative Exponent Rule, the equation should be 10 ^ (- 1/3) = - 1/(root(10, 3)) True, because the Negative Exponent Rule requires that the number be rewritten as a negative value.

Answer 1

To determine whether the equation \( 10^{(-1/3)} = -\sqrt[3]{10} \) is true or false, we can apply the properties of exponents and roots.

First, we rewrite \( 10^{(-1/3)} \) using the Negative Exponent Rule:

\[ 10^{(-1/3)} = \frac{1}{10^{1/3}} \]

Next, recognize that \( 10^{1/3} \) can be rewritten as the cube root of \( 10 \):

\[ 10^{1/3} = \sqrt[3]{10} \]

So, we have:

\[ 10^{(-1/3)} = \frac{1}{\sqrt[3]{10}} \]

Now, we need to compare \( \frac{1}{\sqrt[3]{10}} \) and \(-\sqrt[3]{10}\):

• \( \frac{1}{\sqrt[3]{10}} \) is a positive quantity, since the cube root of a positive number is positive.
• \(-\sqrt[3]{10}\) is negative because it has a negative sign in front.

Since the left side is positive and the right side is negative, the equation \( 10^{(-1/3)} = -\sqrt[3]{10} \) is false.

Thus, the correct response is:

False, because according to the Negative Exponent Rule, the equation should be \( 10^{(-1/3)} = \frac{1}{\sqrt[3]{10}} \), and this cannot equal \(-\sqrt[3]{10}\).