Questions: HW 10.1 Question 5, 10.1.21 Find the cube root. ∛(-1/729) Select the correct choice below and, if necessary, fill in the answer box within your choice A. ∛(-1/729)= B. The root is not a real number.

Solution Steps

To find the cube root of a number, we need to determine the value that, when multiplied by itself three times, equals the original number. In this case, we are looking for the cube root of $-\frac{1}{729}$. Since the cube root of a negative number is also negative, we can calculate the cube root directly.

To find the cube root of $-\frac{1}{729}$, we express it mathematically as: $$\sqrt[3]{-\frac{1}{729}} = \left(-\frac{1}{729}\right)^{\frac{1}{3}}$$

The expression can be simplified as follows: $$\left(-\frac{1}{729}\right)^{\frac{1}{3}} = -\left(\frac{1}{729}\right)^{\frac{1}{3}} = -\frac{1}{9}$$

The cube root of $-\frac{1}{729}$ is a real number, specifically: $$\sqrt[3]{-\frac{1}{729}} = -\frac{1}{9}$$

Final Answer