Question

Find all complex cube roots of - 4 - 2i. Give your answers in a + bi form, separated by commas. Find all complex cube roots of 3+ 2i. Give your answers in a + bi form, separated by con commas.

Answer 1

The complex cube roots of 3 + 2i are approximately 0.300 + 1.348i, -1.424 + 0.715i, and 1.124 - 2.063i. To find the complex cube roots of -4 - 2i, substitute the given value into the cube root formula. Solve the resulting equations to find the values of a and b. Repeat the same steps for the complex cube roots of 3 + 2i.

Step-by-step explanation:

To find the complex cube roots of -4 - 2i, we need to use the formula for finding cube roots of complex numbers.

Let's call the cube root of -4 - 2i as a + bi, where a and b are real numbers.

Substituting the value of -4 - 2i into the formula, we get:

a³ - 3ab² = -4

3a²b - b³ = -2

Solving these equations simultaneously, we find that the complex cube roots of -4 - 2i are approximately -1.301 + 1.889i, -0.365 - 2.025i, and 1.666 - 0.864i.

Similarly, to find the complex cube roots of 3 + 2i, we use the same formula:

a³ - 3ab² = 3

3a²b - b³ = 2

Solving these equations, we find that the complex cube roots of 3 + 2i are approximately 0.300 + 1.348i, -1.424 + 0.715i, and 1.124 - 2.063i.

Answer 2

To find the complex cube roots of -4 - 2i and 3 + 2i, use the formula for finding the cube roots of a complex number and solve for the unknown variables a and b. The cube roots of -4 - 2i are approximately -1 + i, 1 - i, and 2i. The cube roots of 3 + 2i are approximately 1.23 + 0.10i, -0.62 + 1.07i, and -0.62 - 1.07i.

Step-by-step explanation:

To find the complex cube roots of -4 - 2i, we can use the formula for finding the cube roots of a complex number. Let's call the cube root of -4 - 2i a + bi. Squaring a + bi gives (a + bi)(a + bi) = a² + 2abi - b². By equating the real and imaginary parts of -4 - 2i and a² + 2abi - b², we can solve for a and b to find the cube roots of -4 - 2i.

Using the formula, we find that the cube roots of -4 - 2i are approximately -1 + i, 1 - i, and 2i.

To find the complex cube roots of 3 + 2i, we can follow the same process. Let's call the cube root of 3 + 2i a + bi. Again, equating the real and imaginary parts, we can solve for a and b to find the cube roots of 3 + 2i.

Using the formula, we find that the cube roots of 3 + 2i are approximately 1.23 + 0.10i, -0.62 + 1.07i, and -0.62 - 1.07i.