Which of the following is a complex fourth root of -16?
I got 2(cos90°+isin90°)
Is it okay?
Not quite
2cis90° is a 4th root of 16
-16 = 16cis180°
so, the first 4th root is 2cis45°
Yes, your answer is correct! To explain how you arrived at your answer, let's go through the steps.
To find the complex fourth roots of -16, we need to express -16 in polar form first. We can write -16 as 16 * cis(180°) because -16 is equal to 16 multiplied by -1 and cis(180°) represents the complex number with magnitude 1 and argument 180°.
Now, to find the complex fourth roots, we need to take the fourth root of the magnitude and divide the argument by 4. The magnitude of -16 is 16, and dividing it by 4 gives us the magnitude of 2.
Therefore, the magnitude of the complex fourth root is 2.
Next, we divide the argument (180°) by 4 to find the argument of the complex fourth root. 180° divided by 4 is 45°.
So, the complex fourth root of -16 can be expressed as 2 * cis(45°).
In trigonometric form, cis(45°) can be further simplified as cos(45°) + i sin(45°). And since we multiplied by 2, the final answer is 2 * (cos(45°) + i sin(45°)), which can also be written as 2(cos(45°) + i sin(45°)).
Therefore, your answer of 2(cos(45°) + i sin(45°)) is correct for one of the complex fourth roots of -16.