write the following numbers in a +bi form

Question

A. (4+4i)/(-4+4i)=

B. -3/3i + -4/2i=

C. (5i)^3 =

Answer 1

A

multiply both numerator and denominator by the conjugate of the denominator

conjugate of the denominator is 4+4i

(4+4i)/(-4+4i) = (16+32i -16)/(-16-16)= 32i/-32 = -i

-i in a+bi form would be = 0-1i

B is

-3/3i + -4/2i = -1/i - 2/i = -3/i = -3i/i^2 = -3i/-1 = 3i = 0+3i

C (5i)^3 = (5)^3(i)^3 = 125(-i) = -125i = 0 -125i

Answer 2

A) This one you can solve by multiplying the top and bottom by the conjugate of the bottom. That is, in the bottom pair, you flip the sign of the imaginary component, and then multiply the top and bottom each by that.

(4+4i)/(-4+4i) = (4+4i) * (-4-4i) / ((-4+4i)*(-4-4i))

doing this changes the complex number in the denominator to a real number.

(4+4i) * (-4-4i) / ((-4+4i)*(-4-4i)) = (4*(-4) + 4*(-4i) + 4i*(-4) + 4i*(-4i)) / ((-4)² + (-4)*(-4i) + 4i*(-4) + 4i*(-4i))

= (-16 -16i -16i -16i²) / (16 + 16i - 16i - 16i²) = (-16 - 32i -16i²) / (16 - 16i²)

finally, we can use the fact that i² = -1 to give us:

(0 - 32i) / (32) = 0 - 1i.

For part B, you can get started by multiplying the top and bottom of each fraction in such a way that they both share a common denominator (hint: this will be 6i), and then use what you know about fractions and a certain identity of 1/i to get you to your answer.

For part C, remember that when a product is raised to a power, you can split them like this: (ab)^x = a^xb^x. The rest of the problem should be nice and easy from there.

If you're still struggling with these problems or more like them, I encourage you to reach out, and we can talk about scheduling some tutoring sessions! My door is [pretty much] always open. Best!

-Miles