Solve 3+5i/7+i

$$\frac { 3 + 5 i } { 7 + i }$$

Evaluate

$$\frac{13}{25}+\frac{16}{25}i=0.52+0.64i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 7-i.

$$\frac{\left(3+5i\right)\left(7-i\right)}{\left(7+i\right)\left(7-i\right)}$$

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

$$\frac{\left(3+5i\right)\left(7-i\right)}{7^{2}-i^{2}}$$

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

$$\frac{\left(3+5i\right)\left(7-i\right)}{50}$$

Multiply complex numbers 3+5i and 7-i like you multiply binomials.

$$\frac{3\times 7+3\left(-i\right)+5i\times 7+5\left(-1\right)i^{2}}{50}$$

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

$$\frac{3\times 7+3\left(-i\right)+5i\times 7+5\left(-1\right)\left(-1\right)}{50}$$

Do the multiplications in 3\times 7+3\left(-i\right)+5i\times 7+5\left(-1\right)\left(-1\right).

$$\frac{21-3i+35i+5}{50}$$

Combine the real and imaginary parts in 21-3i+35i+5.

$$\frac{21+5+\left(-3+35\right)i}{50}$$

Do the additions in 21+5+\left(-3+35\right)i.

$$\frac{26+32i}{50}$$

Divide 26+32i by 50 to get \frac{13}{25}+\frac{16}{25}i.

$$\frac{13}{25}+\frac{16}{25}i$$

Real Part

$$\frac{13}{25} = 0.52$$

Solution Steps

Multiply both numerator and denominator of \frac{3+5i}{7+i} by the complex conjugate of the denominator, 7-i.

$$Re(\frac{\left(3+5i\right)\left(7-i\right)}{\left(7+i\right)\left(7-i\right)})$$

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

$$Re(\frac{\left(3+5i\right)\left(7-i\right)}{7^{2}-i^{2}})$$

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

$$Re(\frac{\left(3+5i\right)\left(7-i\right)}{50})$$

Multiply complex numbers 3+5i and 7-i like you multiply binomials.

$$Re(\frac{3\times 7+3\left(-i\right)+5i\times 7+5\left(-1\right)i^{2}}{50})$$

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

$$Re(\frac{3\times 7+3\left(-i\right)+5i\times 7+5\left(-1\right)\left(-1\right)}{50})$$

Do the multiplications in 3\times 7+3\left(-i\right)+5i\times 7+5\left(-1\right)\left(-1\right).

$$Re(\frac{21-3i+35i+5}{50})$$

Combine the real and imaginary parts in 21-3i+35i+5.

$$Re(\frac{21+5+\left(-3+35\right)i}{50})$$

Do the additions in 21+5+\left(-3+35\right)i.

$$Re(\frac{26+32i}{50})$$

Divide 26+32i by 50 to get \frac{13}{25}+\frac{16}{25}i.

$$Re(\frac{13}{25}+\frac{16}{25}i)$$

The real part of \frac{13}{25}+\frac{16}{25}i is \frac{13}{25}.

$$\frac{13}{25}$$