Calculating the Imaginary Part of (1-2i)^2-i

[Anood]

How can i find The imaginary part of (1-2i)^2-i.

This is what i have done so far:
(1-2i)^2 (1-2i)^-i

=(1-4i+4)(1-2i)^-i

 

[neutrino]

Use the polar form of 1-2i.

 

[tacman]

=5(1-2i)^-i

To find the imaginary part of (1-2i)^-i, we can use the following steps:

1. Rewrite (1-2i)^-i as (1-2i)^-1 * (1-2i)^-i. This will allow us to use the power rule for exponents.

2. Simplify (1-2i)^-1 as 1/(1-2i).

3. Rewrite (1-2i)^-i as (1-2i)^-1 * i^-1. This will allow us to use the power rule for exponents again.

4. Simplify i^-1 as -i.

5. Substitute these simplified values back into the original expression: 5(1-2i)^-i = 5(1-2i)^-1 * (-i) = 5(1/(1-2i)) * (-i)

6. Multiply the complex numbers in the denominator using the FOIL method: 5(1/(1-2i)) * (-i) = 5(1-2i)/(1-2i) = 5i/(1-2i)

7. Simplify this expression by rationalizing the denominator: 5i/(1-2i) = 5i(1+2i)/(1-4i^2) = 5i(1+2i)/(1+4) = 5i(1+2i)/5 = i(1+2i)

8. Finally, we can see that the imaginary part of (1-2i)^2-i is 2i.