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

2 Answers
Steve M
Sep 8, 2017

# (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#

Taha Z.
Sep 8, 2017

#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.

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