Use the properties of exponents to determine whether the equation 10 -1/3 = 3√ is true or false. Select the response that correctly answers the question and provides the appropriate justification.

Question

Responses

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.

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 = 1/3√10.

Answer 1

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

1. The left side \(10^{-1/3}\) represents \(1 / 10^{1/3}\) according to the Negative Exponent Rule. This is equivalent to \(1 / \sqrt[3]{10}\).

2. The right side \(3\sqrt{10}\) is simply \(3 \times 10^{1/2}\).

Now, comparing both sides:

• Left side: \(10^{-1/3} = \frac{1}{\sqrt[3]{10}}\)
• Right side: \(3\sqrt{10} = 3 \times 10^{1/2} = 3 \times \sqrt[2]{10}\)

Since \(10^{-1/3}\) is much smaller than \(3\sqrt{10}\) (because the left represents a fraction while the right side is multiplied by 3), the equation is not true.

Thus, the correct response is:

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