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\frac{\left(1+3i\right)\left(3+2i\right)}{\left(3-2i\right)\left(3+2i\right)}
Multiply both numerator and denominator by the complex conjugate of the denominator, 3+2i.
\frac{\left(1+3i\right)\left(3+2i\right)}{3^{2}-2^{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{\left(1+3i\right)\left(3+2i\right)}{13}
By definition, i^{2} is -1. Calculate the denominator.
\frac{1\times 3+1\times \left(2i\right)+3i\times 3+3\times 2i^{2}}{13}
Multiply complex numbers 1+3i and 3+2i like you multiply binomials.
\frac{1\times 3+1\times \left(2i\right)+3i\times 3+3\times 2\left(-1\right)}{13}
By definition, i^{2} is -1.
\frac{3+2i+9i-6}{13}
Do the multiplications in 1\times 3+1\times \left(2i\right)+3i\times 3+3\times 2\left(-1\right).
\frac{3-6+\left(2+9\right)i}{13}
Combine the real and imaginary parts in 3+2i+9i-6.
\frac{-3+11i}{13}
Do the additions in 3-6+\left(2+9\right)i.
-\frac{3}{13}+\frac{11}{13}i
Divide -3+11i by 13 to get -\frac{3}{13}+\frac{11}{13}i.
Re(\frac{\left(1+3i\right)\left(3+2i\right)}{\left(3-2i\right)\left(3+2i\right)})
Multiply both numerator and denominator of \frac{1+3i}{3-2i} by the complex conjugate of the denominator, 3+2i.
Re(\frac{\left(1+3i\right)\left(3+2i\right)}{3^{2}-2^{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{\left(1+3i\right)\left(3+2i\right)}{13})
By definition, i^{2} is -1. Calculate the denominator.
Re(\frac{1\times 3+1\times \left(2i\right)+3i\times 3+3\times 2i^{2}}{13})
Multiply complex numbers 1+3i and 3+2i like you multiply binomials.
Re(\frac{1\times 3+1\times \left(2i\right)+3i\times 3+3\times 2\left(-1\right)}{13})
By definition, i^{2} is -1.
Re(\frac{3+2i+9i-6}{13})
Do the multiplications in 1\times 3+1\times \left(2i\right)+3i\times 3+3\times 2\left(-1\right).
Re(\frac{3-6+\left(2+9\right)i}{13})
Combine the real and imaginary parts in 3+2i+9i-6.
Re(\frac{-3+11i}{13})
Do the additions in 3-6+\left(2+9\right)i.
Re(-\frac{3}{13}+\frac{11}{13}i)
Divide -3+11i by 13 to get -\frac{3}{13}+\frac{11}{13}i.
-\frac{3}{13}
The real part of -\frac{3}{13}+\frac{11}{13}i is -\frac{3}{13}.