Question

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.

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 \(3\sqrt{10}\).**