Let z=−93√−9i.
When z is graphed in the complex plane, for what values of n does zn lie on the positive x-axis?
To find the values of n for which zn lies on the positive x-axis when z = -93√(-9i), we need to convert z to polar form and examine its argument.
Let's start by finding the polar form of z. The formula for converting from rectangular to polar form is:
z = r(cosθ + isinθ),
where r is the magnitude of z and θ is the argument of z.
To find the magnitude of z, we use the formula:
r = |z| = √(Re(z)² + Im(z)²),
where Re(z) is the real part of z and Im(z) is the imaginary part of z.
Given z = -93√(-9i), we can rewrite it as:
z = -93 * √(-9) * √i.
√(-9) is equal to 3i, so we have:
z = -93 * 3i * √i.
Now, let's calculate the magnitude (r) of z:
r = |z| = √(Re(z)² + Im(z)²).
Re(z) = 0 since z does not have a real part.
Im(z) = -93 * 3 = -279.
Therefore:
r = |z| = √(0 + (-279)²) = 279.
To find the argument (θ) of z, we use the formula:
θ = arctan(Im(z) / Re(z)).
In this case, since Re(z) is 0, the argument is undefined.
The polar form of z can be written as:
z = r * (cosθ + isinθ) = 279 * (cosθ + isinθ).
Now, let's consider zn. We know that z lies on the positive x-axis when its argument is a multiple of 180 degrees or π radians.
The argument of zn can be found using De Moivre's theorem:
θn = n * θ.
Since the argument of z is undefined (θ = undefined), the argument of zn will also be undefined for any non-zero integer value of n. Therefore, there are no values of n for which zn lies on the positive x-axis.
In conclusion, when z is graphed in the complex plane, there are no values of n for which zn lies on the positive x-axis.