Which of the following is the correct answer for (4+2i)^4 evaluated and left in Cartesian form?

Question

1. Pick the correct answer for the evaluation of (4+2i)⁴, leaving the answer in Cartesian form. Show each step taken in finding the solution. The possible answers are: (A) -112+384i, (B) -120+394i, (C) -123+418i, (D) -88+452i, and (E) None of the Above.
2. Find the answer, in polar form, of the expression 3√125e^13π/10i.
3. Put -x²+8x+8y=64 into standard form, filling in the blank with the answer: (x-4)² = ______________.
4. John is constructing a satellite dish. The receiver is going to be located 15 inches above the vertex of the dish. The dish is going to be 72 inches wide. How deep will the dish be? The possible answers are (A) 45.6 inches, (B) 37.6 inches, (C) 1.6 inches, (D) 21.6 inches, and (E) None of the Above.

Answer 1

I agree with Paul M. I'm available the rest of the day if you'd like to Zoom. Getting others to do your work for you is not the purpose of this forum.

Answer 2

Problem 1

z = 4+2i

r = √(a²+b²) = √(4²+2²) = √(16+4) = √20 = 2√5

θ = tan^-1(b/a) = tan^-1(2/4) = tan^-1(1/2) ≈ 0.464 <-- in radians

zⁿ = rⁿ(cos(nθ)+i·sin(nθ)) <-- DeMoivre's Theorem

z⁴ = (2√5)⁴(cos(4(0.464))+i·sin(4(0.464)))

z⁴ = 400(cos(1.856)+i·sin(1.856))

z⁴ = 400(-0.28+0.96i)

z⁴ = -112+384i <-- Option A is correct

Problem 2

³√125e^(13π/10)i

= 5e^(13π/10)i

= 5(cos(13π/10)+i·sin(13π/10))

= 5(-0.59-0.81i)

= -2.95-4.05i <-- Full simplification if needed

Problem 3

-x²+8x+8y = 64

x²-8x-8y = -64

(x²-8x)-8y = -64

(x²-8x+16)-8y = -64+16

(x-4)²-8y = -48

(x-4)² = 8y-48

Problem 4

The receiver of a satellite dish is generally located at the focus. Recall the equation x²=4ay where "a" is the distance from the focus to the vertex. Given the receiver is going to be located 15 inches above the vertex of the dish, then the distance between the vertex and the focus will be a=15, so now we'll need to solve for "y" which is the depth of the dish:

x² = 4ay

36² = 4(15)y

1296 = 60y

21.6 = y

Therefore, the satellite dish will be 21.6 inches deep, making option D correct!