How do you simplify (7+24i)^.5 ?

Question

These dumb little answers that are just earning you points aren't helping me at all, folks. This isn't my homework. I'm not taking a math course this semester. I'm trying to help a friend who IS in math, but has never used Yahoo Answers. I would appreciate it if you would just stop telling me what I already know, and explain this problem to me. Thanks.

Thanks for the real answers. They are all very helpful.

Answer 1

For your reference:

First note that e^(iθ) = cosθ + isinθ

Any solution to z^n = a where z and a are complex numbers and n is an integer >=0 looks like:

z = |a|^(1/n) * e^[(i/n)(θ+2πk)] for k = 0, ..., n-1

where θ = Arg(a).

So in this problem: z = (7+24i)^.5

This means that
z^2 = 7 + 24i

I will assume all angles are in radians.

Notice the following:
θ = Arg(7+24i) = arctan(24/7) = 1.287
|a| = |7+24i| = Sqrt(7^2 + 24^2) = Sqrt(625) = 25
n = 2
k = 0, 1

Substituting, we get two answers:

z_1 = 25^(1/2) * e^[(i/2)(1.287+2π(0))]
= 5 * e^[i(0.644)]
= 5 * (cos(0.644)+isin(0.644))
= 5 * (0.800 + i(0.600))
= 5 * (4/5 + i(3/5))
= 4 + 3i

z_2 = 25^(1/2) * e^[(i/2)(1.287+2π(1))]
= 5 * e^[i(3.785)]
= 5 * (cos(3.785)+isin(3.785))
= 5 * (-0.800 + i(-0.600))
= 5 * (-4/5 - i(3/5))
= -4 - 3i

Answer 2

1. Start by simplifying 7+24i. This is in the form x + iy. You can
convert this to phasor form, Re^(i*theta), where
R = SQRT((x^2)+(y^2)), and theta = arctan(y/x). Try it. you
should get 25*e^(i*arctan(24/7)).

2. Now take the square root of the above, and you have the
answers, +/- 5*e^(i/2*arctan(24/7)).

Answer 3

What you are really asking is how do you take the square root of a complex number?

Let me start with a + bi then square it to get
(a + bi)^2 = (a^2-b^2) + abi

so now you can use this to find the square root, by solving

(a^2 - b^2) = 7 and ab = 24

Once you solve for a and b then you will have the square root of your equation as a + bi.

This is one way to solve your problem. There may be others.

Another way is to use this website

http://www.alpertron.com.ar/CALC.HTM

Answer 4

(7+24i)^.5 is the same as square root(7+24i).. i put it into my calculator cuz i hate i and i get 4+3i...hope thats rite..

Answer 5

No one is going to do your homework for you :)

Answer 6

do it on the calculator