Use Demoivre's theorem to find the indicated power of the complex number. Write the result in standard form.

1.(3(cos5pi/4+i sin 5pi/4))^8
2.(3-3i)^6

(3(cos(5pi/4)+i sin (5pi/4) ) )^8

= 3^8(cos 8(5π/4) + i sin 8(5π/4))
= 6561( cos 10π + i sin 10π)
= 6561( 1 + i(0) )
= 6561

B
(3-3i)^6
let z = 3-3i, so we want z^6

r = √(9 + 9) = √18 = 3√2
plot (3,-3) which is in IV, so
Ø = 315° or 7π/4
z = 3√2(cos 7π/4 + i sin 7π/4)
z^6 = (3√2)^6 [cos 42π/4 + i sin 42π/4]
= 5832( 0 + i (1) )
= 5832 i

To use De Moivre's Theorem to find the indicated power of a complex number, we can follow these steps:

Step 1: Express the complex number in polar form.
Step 2: Apply De Moivre's Theorem.
Step 3: Convert the result back to rectangular or standard form.

Let's solve the given problems step by step:

1. (3(cos(5π/4) + i sin(5π/4)))^8

Step 1: Express the complex number in polar form.
To express a complex number in polar form, we can use the formula:

r(cosθ + i sinθ)

In this case, the complex number is already given in polar form, with a magnitude of 3 (r = 3) and an angle of 5π/4 (θ = 5π/4).

Step 2: Apply De Moivre's Theorem.
De Moivre's Theorem states that for any complex number z in polar form (r(cosθ + i sinθ)), and any integer n:

z^n = r^n (cos(nθ) + i sin(nθ))

In our case, z = 3(cos(5π/4) + i sin(5π/4)) and n = 8. Plugging these values into the theorem:

z^8 = (3^8) (cos(8 * 5π/4) + i sin(8 * 5π/4))

Simplifying:
z^8 = 6561 (cos(10π) + i sin(10π))

Step 3: Convert the result back to rectangular or standard form.
To convert the result back to rectangular form, we can simplify the trigonometric equation:

cos(10π) = cos(2π * 5) = cos(0) = 1
sin(10π) = sin(2π * 5) = sin(0) = 0

Therefore:
z^8 = 6561 (1 + i * 0)
z^8 = 6561

The result in standard form is simply 6561.

2. (3 - 3i)^6

Step 1: Express the complex number in polar form.
To express a complex number in polar form, we can use the formula:

r(cosθ + i sinθ)

In this case, the complex number is given in rectangular form, with a real part of 3 and an imaginary part of -3.

To convert it to polar form, we need to find the magnitude (r) and the angle (θ).

Magnitude, r:
r = √(a^2 + b^2)
r = √(3^2 + (-3)^2)
r = √(9 + 9)
r = √18 = 3√2

Angle, θ:
θ = arctan(b/a)
θ = arctan(-3/3)
θ = arctan(-1)
θ = -π/4

Expressing the complex number in polar form:
3 - 3i = 3√2(cos(-π/4) + i sin(-π/4))

Step 2: Apply De Moivre's Theorem.
We can apply De Moivre's Theorem as before:
(3 - 3i)^6 = (3√2)^6 (cos(6 * -π/4) + i sin(6 * -π/4))

Simplifying:
(3 - 3i)^6 = 729 (cos(-3π/2) + i sin(-3π/2))

Step 3: Convert the result back to rectangular or standard form.
To convert the result back to rectangular form, we can simplify the trigonometric equation:

cos(-3π/2) = cos(-π/2) = 0
sin(-3π/2) = sin(-π/2) = -1

Therefore:
(3 - 3i)^6 = 729 (0 - i)
(3 - 3i)^6 = -729i

The result in standard form is -729i.