a) 1/z in polar form is √2/2 * e^(i45°).
b) z¹⁰ in polar form is 32 * e^(-i90°).
c) z⁸.z* in polar form is 16.
The solutions to the given problems:
a) 1/Z
Given a complex number z = 1-j, we can find its reciprocal in polar form using the following formula:
1/z = 1/r * e^(-iθ)
where r is the absolute value of z and θ is the angle of z.
First, let's find the absolute value of z:
|z| = |1-j| = √(1² + (-1)²) = √2
Next, let's find the angle of z:
θ = arctan(Im(z)/Re(z)) = arctan(-1/1) = -45°
Now, we can plug these values into the formula to find 1/z in polar form:
1/z = 1/√2 * e^(-i(-45°)) = √2/2 * e^(i45°)
Therefore, 1/z in polar form is √2/2 * e^(i45°).
b) z¹⁰
To find z¹⁰ in polar form, we can use the following formula:
z^n = r^n * e^(inθ)
where n is the power of z and r and θ are the absolute value and angle of z, respectively.
We already found that r = √2 and θ = -45°. Plugging these values into the formula, we get:
z¹⁰ = (√2)^10 * e^(i(-45°)(10)) = 2^(5) * e^(i(-450°)) = 32 * e^(-i90°)
Therefore, z¹⁰ in polar form is 32 * e^(-i90°).
c) z⁸.z*
To find z⁸.z* in polar form, we can use the following formula:
(z^n)(z*) = r^n * e^(inθ) * r * e^(-iθ) = r^(n+1)
where n is the power of z, r and θ are the absolute value and angle of z, respectively, and z* is the complex conjugate of z.
We already found that r = √2. Plugging this value into the formula, we get:
(z^8)(z*) = (√2)^(8+1) = 2^(4) = 16
Therefore, z⁸.z* in polar form is 16.