Write each expression in problems 1 to 3 in the form a + bi.
1. (8-3i)+(-6+2i)
2. 4/(2-i)
3. (3-2i)^3
I do not understand at all where to start. I tired looking on google to find a way to solve but I didn't get any real help. So please help me!
1. 8 - 6 -3i + 2i = 2 - i.
2. 4 / (2 - i),
Multiply numerator and denominator by 2 + i:
(8 + 4i) / (2 - i)(2 + i),
(8 + 4i) / (4 + 1) = (8 + 4i) / 5.
3. (3 - 2i)^3 = (3 - 2i)^2 (3 - 2i)=
(9 - 12i + 4) (3 - 2i),
(13 - 12i) (3 - 2i),
39 - 26i - 36i - 24 = 15 - 62i.
Alright thank you. Can I also put them in my calculator? I wasn't sure but I did for the first one and got the same answer.
That's fine if you don't have to show
your work.
Thank you so much for your help!
To write the expressions in the form a + bi, we need to simplify the given expressions and separate the real and imaginary parts.
Let's go through each problem step by step:
1. (8-3i)+(-6+2i):
To simplify this expression, we'll combine the real and imaginary parts separately.
Real part: (8-3i) + (-6+2i) = 8 - 6 = 2
Imaginary part: (-3i) + (2i) = -3i + 2i = -1i
So, the expression (8-3i)+(-6+2i) can be written in the form a + bi as 2 - i.
2. 4/(2-i):
To write this expression in the form a + bi, we'll rationalize the denominator (multiply both the numerator and denominator by the conjugate of the denominator).
Conjugate of (2-i) = (2+i)
Numerator: 4 * (2+i) = 8 + 4i
Denominator: (2-i) * (2+i) = 4 - i^2 = 4 + 1 = 5
So, the expression 4/(2-i) can be written as (8 + 4i)/5.
3. (3-2i)^3:
We'll expand this expression using the binomial theorem.
(3-2i)^3 = (3-2i)(3-2i)(3-2i)
Expanding this, we get:
= (9 - 6i - 6i + 4i^2)(3 - 2i)
= (9 - 12i + 4i^2)(3 - 2i)
= (9 - 12i - 4)(3 - 2i)
= (5 - 12i)(3 - 2i)
= 5 * 3 + 5 * (-2i) - 12i * 3 - 12i * (-2i)
= 15 - 10i - 36i + 24i^2
= 15 - 10i - 36i + 24(-1)
= 15 - 10i - 36i - 24
= -9 - 46i
So, the expression (3-2i)^3 can be written as -9 - 46i.
I hope this helps! Let me know if you have any further questions.