Hello there can someone please help me Rationalizing Imaginary Denominators (Simplify)

Question

(–4+10i)/(3+6i) 

Answer 1

When you are rationalizing a denominator like this you multiply the top and bottom by the denominator's conjugate.

So, what's a conjugate?

I am glad you asked.

A conjugate will have the same terms as the denominator, but the opposite operator between the two terms.

Since the denominator is 3 + 6i, the conjugate is 3 - 6i. So, let's multiply the top and bottom by 3 - 6i.

Since this format is not fraction friendly, let's multiply the numerator by 3 - 6i, then we will multiply the denominator by 3 - 6i, and finally we will put everything together.

Numerator

(-4 + 10i)(3 - 6i) = -12 + 24i + 30i - 60i² FOIL

= -12 + 54i - 60i² combine like terms

= -12 + 54i - 60(-1) when simplifying always replace i² with -1

= -12 + 54i +60 combine like terms again

= 48 + 54i Whew! It is in a + bi form.

Denominator

(3 + 6i)(3 - 6i) = 9 -18i + 18i - 36i² FOIL

= 9 - 36i² Combine like terms. Notice the "i" terms added to zero.

This will always happen when multiplying conjugates.

= 9 - 36(-1) when simplifying always replace i² with -1

= 9 + 36 = 45 Yay! The denominator is real!!!

Put the Numerator and Denominator together

(48 + 54i)/45 Your teacher probably wants this in a + bi form

48/45 + (54/45)i Reduce each fraction.

16/15 + (6/5)i Here is your answer in a + bi form.

If your teacher does not require a + bi form, you can simply reduce (48 + 54i)/45 by 3 since all terms have 3 as a factor. Your final answer would be (16 + 18i)/15.

Answer 2

Any time that you have rational complex expressions, the best approach is to multiply the denominator by its complex conjugate to simply. So for your given problem:

(–4+10i)/(3+6i)  = (-4+10i)(3-6i)/(3+6i)(3-6i) = (-12 + 24i +30i +60)/(9 + 36) = (48 + 54i)/45.