How do you simplify #1/(2+5i) #?

1 Answer
Jan 31, 2016

Answer:

To simplify a complex fraction, multiply top and bottom (denominator and numerator) by the complex conjugate of the bottom. This expression simplifies to #(2-5i)/29#.

Explanation:

The 'complex conjugate' of a complex number #(a+bi)# is #(a-bi)#, and multiplying a complex number by its complex conjugate will yield a real number.

#1/(2+5i)*(2-5i)/(2-5i)#

Note that

#(2-5i)/(2-5i) = 1# so we are not changing the original number by simplifying it.

#1/(2+5i)*(2-5i)/(2-5i) = (2-5i)/(4+10i-10i-25i^2) = (2-5i)/(4-25i^2)#

Note that #i^2=-1# because #i=sqrt(-1)#

So the simplified version of this expression is

#(2-5i)/(4-(-25)) = (2-5i)/29#