In Exercises 69–76, find all the complex roots. Write roots in rectangular form. If necessary, round to the nearest tenth.The complex fifth roots of 32

Verified step by step guidance

1. Identify the problem as finding the fifth roots of a complex number, which is 32 in this case.
2. Express 32 in polar form. Since 32 is a real number, it can be written as 32 + 0i, which in polar form is 32(cos(0) + i sin(0)).
3. Use De Moivre's Theorem to find the fifth roots. The theorem states that if z = r(cos(θ) + i sin(θ)), then the nth roots are given by r^(1/n)(cos((θ + 2kπ)/n) + i sin((θ + 2kπ)/n)), where k = 0, 1, 2, ..., n-1.
4. Calculate the magnitude of the roots: r^(1/5) = 32^(1/5).
5. Find the angles for each root: θ_k = (0 + 2kπ)/5 for k = 0, 1, 2, 3, 4, and express each root in rectangular form using cos(θ_k) + i sin(θ_k).

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit defined as the square root of -1. Understanding complex numbers is essential for solving problems involving roots, especially when dealing with non-real solutions.

Roots of Complex Numbers

Finding the roots of complex numbers involves determining the values that, when raised to a certain power, yield the original complex number. For example, the nth roots of a complex number can be found using De Moivre's Theorem, which relates the roots to the polar form of the complex number, allowing for easier calculations of angles and magnitudes.

Rectangular Form

Rectangular form refers to expressing complex numbers in the standard a + bi format. When finding roots, it is often necessary to convert from polar form (which uses magnitude and angle) back to rectangular form to provide a clear and conventional representation of the complex roots, making them easier to interpret and use in further calculations.