How can I write sqrt(18 plus 24i) as a complex number?

Answer 1

To take the square root of a complex number:* Write the original complex number in polar notation (absolute value and angle).

* For the square root, the angle will be half the angle. The absolute value will be the square root of that of the original number.

* There is a second square root: the same as above, but add 180 degrees (or pi, if using radians) to the angle.

* If you wish, you can convert the result(s) back to rectangular coordinates, i.e., real number + imaginary number.

Most scientific calculators have functions to convert from rectangular to polar and vice versa.

Answer 2

For 18 + 24i,r^2 = 18^2 + 24^2 = 900 => r = 30and alpha = atan(24/18) => beta = 0.9273 radians, approx

So, for the solution s = sqrt(r) = sqrt(30)

and beta = alpha/2 = 0.4636 radians.

Then sqrt(18 + 24i) = s*cos(beta) + s*sin(beta)*i

= 4.8990 + 2.4495*i

The second square root is - 4.8990 - 2.4495*i

Answer 3

If you represent the number as a magnitude and an angle, then the square root of that number will be the positive square root of the magnitude, and then divide the angle by 2. To get the other square root, add 180° to the angle (or pi radians) to the angle of the answer.
So your number just happens to form a scale of a 3-4-5 right triangle, so it is sqrt(6*(3 + 4i)), or sqrt(30*(0.6 + 0.8i)). The magnitude is 30, and the magnitude of the square root is sqrt(30).
The angle is approx 53°. The angle of the square root is approx 26.5°, and the other angle is 206.5°