How do you find the 6th root of -i?
Well, the 6th root of -i is like finding a needle in a haystack, but let's give it a shot! Hold on tight, because we're about to get a little mathematical. Start by writing -i in polar form: -i = 1 * e^(3πi/2). Now, let's find the 6th root of the magnitude, which is 1, since |-i| = 1. So, the 6th root of 1 is 1. As for the argument, we have to divide 3π/2 by 6, which gives us π/4. So the 6th roots of -i are 1 * e^(iπ/4), 1 * e^(iπ/4 + 2iπ/6), 1 * e^(iπ/4 + 4iπ/6), 1 * e^(iπ/4 + 6iπ/6), 1 * e^(iπ/4 + 8iπ/6), and 1 * e^(iπ/4 + 10iπ/6). Voila! You've found them, though they might still be hiding in the haystack somewhere.
To find the 6th root of -i, we can use the polar form of complex numbers.
Step 1: Express -i in polar form.
The polar form of -i is (-1, -π/2) or (-1, 270°) because -i lies on the negative imaginary axis.
Step 2: Find the magnitude of the 6th root.
The magnitude of the 6th root can be found by taking the 6th root of the magnitude of -i. The magnitude of -i is |r| = 1, so the magnitude of the 6th root is 1^(1/6) = 1.
Step 3: Find the argument of the 6th root.
To find the argument of the 6th root, we divide the argument of -i by 6. The argument of -i is -π/2 or 270°, so dividing by 6 gives us -π/12 or 45°.
Step 4: Express the 6th root in polar form.
We now have the magnitude (1) and argument (-π/12 or 45°) of the 6th root. We can express the 6th root in polar form as (1, -π/12) or (1, 45°).
Therefore, the 6th root of -i is (1, -π/12) or (1, 45°) in polar form.
To find the 6th root of -i, you need to raise -i to the power of 1/6. Let's break down the steps in solving this:
Step 1: Represent -i in polar form.
To work with complex numbers more effectively, it's often helpful to express them in polar form. The polar form of a complex number is given by the equation z = r * (cos(theta) + i*sin(theta)), where r is the magnitude of the complex number and theta is the angle it makes with the positive real axis.
For -i, the magnitude (r) is 1, and the angle (theta) is 270 degrees (or 3π/2 radians). This can be obtained by analyzing the position of -i on the complex plane.
Step 2: Express the 6th root of -i in polar form.
To find the 6th root, we need to divide the angle theta by 6. Thus, the new angle (theta') is (3π/2) / 6 = π/4 radians or 45 degrees.
Step 3: Calculate the magnitude and angle of the 6th root.
Since the magnitude remains the same in this case (r = 1), the new complex number representing the 6th root will have a magnitude of 1 and an angle of π/4 radians (45 degrees).
Step 4: Convert the 6th root back to rectangular form (a + bi).
To convert the polar form back to rectangular form, use the formula z = r * (cos(theta') + i*sin(theta')). Plugging in the values, we have:
z = 1 * (cos(π/4) + i*sin(π/4))
= 1 * (sqrt(2)/2 + i*sqrt(2)/2)
= sqrt(2)/2 + i*sqrt(2)/2
So, the 6th root of -i is sqrt(2)/2 + i*sqrt(2)/2.