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\frac{\left(-3-3i\right)\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{\left(-3-3i\right)\left(-3+i\right)}{\left(-3\right)^{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(-3-3i\right)\left(-3+i\right)}{10}
By definition, i^{2} is -1. Calculate the denominator.
\frac{-3\left(-3\right)-3i-3i\left(-3\right)-3i^{2}}{10}
Multiply complex numbers -3-3i and -3+i like you multiply binomials.
\frac{-3\left(-3\right)-3i-3i\left(-3\right)-3\left(-1\right)}{10}
By definition, i^{2} is -1.
\frac{9-3i+9i+3}{10}
Do the multiplications in -3\left(-3\right)-3i-3i\left(-3\right)-3\left(-1\right).
\frac{9+3+\left(-3+9\right)i}{10}
Combine the real and imaginary parts in 9-3i+9i+3.
\frac{12+6i}{10}
Do the additions in 9+3+\left(-3+9\right)i.
\frac{6}{5}+\frac{3}{5}i
Divide 12+6i by 10 to get \frac{6}{5}+\frac{3}{5}i.
Re(\frac{\left(-3-3i\right)\left(-3+i\right)}{\left(-3-i\right)\left(-3+i\right)})
Multiply both numerator and denominator of \frac{-3-3i}{-3-i} by the complex conjugate of the denominator, -3+i.
Re(\frac{\left(-3-3i\right)\left(-3+i\right)}{\left(-3\right)^{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(-3-3i\right)\left(-3+i\right)}{10})
By definition, i^{2} is -1. Calculate the denominator.
Re(\frac{-3\left(-3\right)-3i-3i\left(-3\right)-3i^{2}}{10})
Multiply complex numbers -3-3i and -3+i like you multiply binomials.
Re(\frac{-3\left(-3\right)-3i-3i\left(-3\right)-3\left(-1\right)}{10})
By definition, i^{2} is -1.
Re(\frac{9-3i+9i+3}{10})
Do the multiplications in -3\left(-3\right)-3i-3i\left(-3\right)-3\left(-1\right).
Re(\frac{9+3+\left(-3+9\right)i}{10})
Combine the real and imaginary parts in 9-3i+9i+3.
Re(\frac{12+6i}{10})
Do the additions in 9+3+\left(-3+9\right)i.
Re(\frac{6}{5}+\frac{3}{5}i)
Divide 12+6i by 10 to get \frac{6}{5}+\frac{3}{5}i.
\frac{6}{5}
The real part of \frac{6}{5}+\frac{3}{5}i is \frac{6}{5}.