Solve 7+9i/3+i

$$\frac { 7 + 9 i } { 3 + i }$$

Evaluate

$$3+2i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 3-i.

$$\frac{\left(7+9i\right)\left(3-i\right)}{\left(3+i\right)\left(3-i\right)}$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(7+9i\right)\left(3-i\right)}{3^{2}-i^{2}}$$

By definition, i^{2} is -1. Calculate the denominator.

$$\frac{\left(7+9i\right)\left(3-i\right)}{10}$$

Multiply complex numbers 7+9i and 3-i like you multiply binomials.

$$\frac{7\times 3+7\left(-i\right)+9i\times 3+9\left(-1\right)i^{2}}{10}$$

By definition, i^{2} is -1.

$$\frac{7\times 3+7\left(-i\right)+9i\times 3+9\left(-1\right)\left(-1\right)}{10}$$

Do the multiplications in 7\times 3+7\left(-i\right)+9i\times 3+9\left(-1\right)\left(-1\right).

$$\frac{21-7i+27i+9}{10}$$

Combine the real and imaginary parts in 21-7i+27i+9.

$$\frac{21+9+\left(-7+27\right)i}{10}$$

Do the additions in 21+9+\left(-7+27\right)i.

$$\frac{30+20i}{10}$$

Divide 30+20i by 10 to get 3+2i.

$$3+2i$$

Real Part

$$3$$

Solution Steps

Multiply both numerator and denominator of \frac{7+9i}{3+i} by the complex conjugate of the denominator, 3-i.

$$Re(\frac{\left(7+9i\right)\left(3-i\right)}{\left(3+i\right)\left(3-i\right)})$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$Re(\frac{\left(7+9i\right)\left(3-i\right)}{3^{2}-i^{2}})$$

By definition, i^{2} is -1. Calculate the denominator.

$$Re(\frac{\left(7+9i\right)\left(3-i\right)}{10})$$

Multiply complex numbers 7+9i and 3-i like you multiply binomials.

$$Re(\frac{7\times 3+7\left(-i\right)+9i\times 3+9\left(-1\right)i^{2}}{10})$$

By definition, i^{2} is -1.

$$Re(\frac{7\times 3+7\left(-i\right)+9i\times 3+9\left(-1\right)\left(-1\right)}{10})$$

Do the multiplications in 7\times 3+7\left(-i\right)+9i\times 3+9\left(-1\right)\left(-1\right).

$$Re(\frac{21-7i+27i+9}{10})$$

Combine the real and imaginary parts in 21-7i+27i+9.

$$Re(\frac{21+9+\left(-7+27\right)i}{10})$$

Do the additions in 21+9+\left(-7+27\right)i.

$$Re(\frac{30+20i}{10})$$

Divide 30+20i by 10 to get 3+2i.

$$Re(3+2i)$$

The real part of 3+2i is 3.

$$3$$