Questions: How many third roots does - 512 have? Explain your reasoning. Choose the correct answer below and, if necessary, fill in the answer boxes to complete your choice. A. There is 1 real root, which is This times itself times equals - 512. B. There are 2 real roots, which are Each of these times themselves (Use a comma to separate answers as needed.) times equals - 512 C. There are 3 real roots, which are Each of these times themselves times equals - 512 (Use a comma to separate answers as needed.) D. There are no real roots. There is no real number that when multiplied by itself any number of times will equal -512.

Solution Steps

To find the third roots of -512, we need to determine the cube roots of the number. A cube root of a number \( x \) is a number \( y \) such that \( y^3 = x \). For negative numbers, there is one real cube root and two complex cube roots. Therefore, -512 has one real cube root and two complex cube roots.

To find the third roots of \( -512 \), we calculate the cube roots of the number. The cube roots can be expressed in the form \( r \cdot e^{i\theta} \), where \( r \) is the magnitude and \( \theta \) is the argument.

The cube roots of \( -512 \) are given by:

1. \( 4 + 6.9282i \)
2. \( -8 + 9.7972 \times 10^{-16}i \) (approximately \( -8 \))
3. \( 4 - 6.9282i \)

Among these roots, we observe that:

• The first root \( 4 + 6.9282i \) and the third root \( 4 - 6.9282i \) are complex.
• The second root \( -8 \) is the only real root.

Final Answer