Evaluate Complex Numbers: \sqrt{\frac{1+j}{4-8j}}

• GreenPrint
In summary, the conversation is about finding the real and complex components of a given number in either rectangular or polar form. The person asking for help is stuck and asks for assistance in using Euler's identity, but realizes that it cannot be used in this case. The responder suggests factoring out a real number to simplify the problem.

Homework Statement

Evaluate (find the real and complex components) of the following numbers, in either rectangular or polar form:

$\sqrt{\frac{1+j}{4-8j}}$

The Attempt at a Solution

I get to here and am not sure where to go from here

$\sqrt{-1/20+3/20j}$

I noticed that I can't used euler's identity here because (-1/20)^2+(3/20)^2 is not one. Thanks for any help you can provide

This would be a good time to change -1/20+3j/20 to polar form.

GreenPrint said:
I noticed that I can't used euler's identity here because (-1/20)^2+(3/20)^2 is not one.
Can you factor out a real number from $-1/20 + 3i/20$ so that what remains has unit magnitude? i.e. write it in the form

$$\frac{-1}{20} + \frac{3i}{20} = r(a + ib)$$

where $a^2 + b^2 = 1$

What is a complex number?

A complex number is a number that contains both a real part and an imaginary part. It is written in the form a + bj, where a is the real part and bj is the imaginary part with b being the imaginary unit.

What is the square root of a complex number?

The square root of a complex number is a number that when multiplied by itself, gives the original complex number. It is written in the form a + bj, where a and b are the square roots of the real and imaginary parts of the original complex number.

Can the square root of a complex number be simplified?

Yes, the square root of a complex number can be simplified by using the rules of complex numbers. For example, in the expression \sqrt{9+16j}, the square root of 9 can be simplified to 3, and the square root of 16 can be simplified to 4, resulting in the simplified expression 3+4j.

How do you evaluate a complex number?

To evaluate a complex number, you need to determine its real and imaginary parts. This can be done by using the rules of complex numbers, such as multiplying and dividing complex numbers, and simplifying square roots. Once the real and imaginary parts are determined, the complex number can be written in the form a + bj.

What is the result of evaluating \sqrt{\frac{1+j}{4-8j}}?

The result of evaluating \sqrt{\frac{1+j}{4-8j}} is \frac{1}{3} + \frac{2}{3}j. This can be simplified by multiplying the numerator and denominator by the conjugate of the denominator, resulting in the expression \frac{1+3j}{13}.