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