Simplify the expression. Type the answer in the standard form of complex numbers . 3/6+i

Answer 1

Explanation:

For fractions, mathematics doesn't like radicals in the denominators, and i is a radical, even if it is imaginary. So to get it out we would multiply by it on the top and the bottom of the fraction, however this time it's a little more complicated. Since the denominator is 6+i it is a complex number (a combination of a real and imaginary number), so we have to multiply by what's called the complex conjugate. The complex conjugate is the exact same numbers, but with a different sign on the imaginary number. In this case 6-i.

Solution:

3/(6+i), we need to multiply 6-i to the top and bottom of the fraction, then reduce where possible.

3*(6-i)=18-3i

(6+i)(6-i)= 36+6i-6i-i²= 36-i²= 36-(-1)= 37

So the result is (18-3i)/37, that can't reduce anywhere because 37 is prime, so final answer

18/37 - 3i/37