Question

Write the expression in rectangular form, x + yi, and in exponential form, rei0
(√6-i)^8

Answer 1

To simplify (√6-i)^8, we can start by expressing the complex number in rectangular form:

Let z = √6 - i = √6 + (-1)i

Then, we can calculate the modulus and argument of z as follows:

|z| = sqrt((√6)^2 + (-1)^2) = sqrt(6 + 1) = sqrt(7)

Arg(z) = arctan(-1/√6) = -0.165148

Now, we can express z in exponential form:

z = |z| * e^(i * Arg(z)) = sqrt(7) * e^(-i * 0.165148)

To raise z to the 8th power, we can use De Moivre's theorem:

z^8 = (sqrt(7))^8 * e^(-8i * 0.165148) = 49 * e^(-1.321184i)

Finally, we can express the result in rectangular form:

z^8 = 49(cos(-1.321184) + i*sin(-1.321184))

z^8 = -31.698 + (-40.844)i