I have to rationalize the denominator of 2+3i/1+2i. I multiplyed this expression by 1-2i/1-2i, resulting in 2+4i+3i+-6i^2/1+-2i+2i+-4i^2. It then became 2+2i+3i+6/1-1+2i+3. Is this right? Thanks!
After multiplying I got
(2 - 4i + 3i - 6i^2)/(1 - 4i^2)
= 2 -i + 6)/(1 + 4)
= 8-i)/5
To rationalize the denominator of a complex number fraction, you need to multiply both the numerator and denominator by the complex conjugate of the denominator.
Let's take a closer look at your steps:
You start with the fraction 2 + 3i / 1 + 2i.
To rationalize the denominator, you multiply both the numerator and denominator by the complex conjugate of the denominator, which is 1 - 2i.
So, the expression becomes:
(2 + 3i) * (1 - 2i) / (1 + 2i) * (1 - 2i)
Now, let's simplify this expression step by step:
Numerator:
2 * 1 = 2
2 * -2i = -4i
3i * 1 = 3i
3i * -2i = -6i^2
Simplifying the numerator, you have: 2 - 4i + 3i - 6i^2.
Next, let's simplify the denominator:
(1 * 1) + (2i * -2i)
Simplifying the denominator, you get: 1 - 4i^2.
Now, let's substitute the value of i^2:
i^2 is equal to -1. So, you have: 2 - 4i + 3i + 6 / 1 - 4(-1).
Simplifying further:
2 - 4i + 3i + 6 / 1 + 4 = 8 - i / 5.
Therefore, the fully simplified rationalized expression is (8 - i) / 5.
In conclusion, your steps are not entirely correct. The correct expression after rationalizing the denominator should be (8 - i) / 5.