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\frac{1}{3+i}
Subtract 1 from 2 to get 1.
\frac{1\left(3-i\right)}{\left(3+i\right)\left(3-i\right)}
Multiply both numerator and denominator by the complex conjugate of the denominator, 3-i.
\frac{1\left(3-i\right)}{3^{2}-i^{2}}
Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.
\frac{1\left(3-i\right)}{10}
By definition, i^{2} is -1. Calculate the denominator.
\frac{3-i}{10}
Multiply 1 and 3-i to get 3-i.
\frac{3}{10}-\frac{1}{10}i
Divide 3-i by 10 to get \frac{3}{10}-\frac{1}{10}i.
Re(\frac{1}{3+i})
Subtract 1 from 2 to get 1.
Re(\frac{1\left(3-i\right)}{\left(3+i\right)\left(3-i\right)})
Multiply both numerator and denominator of \frac{1}{3+i} by the complex conjugate of the denominator, 3-i.
Re(\frac{1\left(3-i\right)}{3^{2}-i^{2}})
Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.
Re(\frac{1\left(3-i\right)}{10})
By definition, i^{2} is -1. Calculate the denominator.
Re(\frac{3-i}{10})
Multiply 1 and 3-i to get 3-i.
Re(\frac{3}{10}-\frac{1}{10}i)
Divide 3-i by 10 to get \frac{3}{10}-\frac{1}{10}i.
\frac{3}{10}
The real part of \frac{3}{10}-\frac{1}{10}i is \frac{3}{10}.