Question

What is 7+6i divided by 10+i?

Answer 1

`(7+6i)/(10+i) = 76/101 + 53/101i`

Explanation:

We can make the denominator real by multiplying the denominator with its complex conjugate, thus:

`(7+6i)/(10+i) = (7+6i)/(10+i) * (10-i)/(10-i)`

`" " = ( (7+6i)(10-i) ) / ( (10+i)(10-i) )`

`" " = ( 70-7i+60i-6i^2 ) / ( 100 -10i + 10i-i^2 )`

`" " = ( 70 + 53i +6 ) / ( 100 +1 )`

`" " = ( 76 + 53i ) / ( 101 )`

`" " = 76/101 + 53/101i`

Answer 2

`76/101 +53/101i`

Explanation:

`(7+6i)/(10+i)`
First we have to rationalize the denominator by multiplying the complex number in the denominator and the numerator by the denominator's conjugate.

`((7+6i)(10-i))/((10+i)(10-i))=(7(10)+6i(10)-7(i)-6i(i))/(10^2-i^2)` (using the difference of squares rule in the denominator)
`=(70+60i-7i-6(i^2))/(100-i^2)=(70+53i-6(-1))/(100-(-1))`**(since `i^2=-1`)

`(70+53i-6(-1))/(100-(-1))=(70+6+53i)/(100+1)=(76+53i)/(101)`

`= 76/101 +53/101i`

I hope that this helps.