Solve 1+5i/7+6i

$$\frac { 1 + 5 i } { 7 + 6 i }$$

Evaluate

$$\frac{37}{85}+\frac{29}{85}i\approx 0.435294118+0.341176471i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 7-6i.

$$\frac{\left(1+5i\right)\left(7-6i\right)}{\left(7+6i\right)\left(7-6i\right)}$$

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

$$\frac{\left(1+5i\right)\left(7-6i\right)}{7^{2}-6^{2}i^{2}}$$

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

$$\frac{\left(1+5i\right)\left(7-6i\right)}{85}$$

Multiply complex numbers 1+5i and 7-6i like you multiply binomials.

$$\frac{1\times 7+1\times \left(-6i\right)+5i\times 7+5\left(-6\right)i^{2}}{85}$$

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

$$\frac{1\times 7+1\times \left(-6i\right)+5i\times 7+5\left(-6\right)\left(-1\right)}{85}$$

Do the multiplications in 1\times 7+1\times \left(-6i\right)+5i\times 7+5\left(-6\right)\left(-1\right).

$$\frac{7-6i+35i+30}{85}$$

Combine the real and imaginary parts in 7-6i+35i+30.

$$\frac{7+30+\left(-6+35\right)i}{85}$$

Do the additions in 7+30+\left(-6+35\right)i.

$$\frac{37+29i}{85}$$

Divide 37+29i by 85 to get \frac{37}{85}+\frac{29}{85}i.

$$\frac{37}{85}+\frac{29}{85}i$$

Real Part

$$\frac{37}{85} = 0.43529411764705883$$

Solution Steps

Multiply both numerator and denominator of \frac{1+5i}{7+6i} by the complex conjugate of the denominator, 7-6i.

$$Re(\frac{\left(1+5i\right)\left(7-6i\right)}{\left(7+6i\right)\left(7-6i\right)})$$

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

$$Re(\frac{\left(1+5i\right)\left(7-6i\right)}{7^{2}-6^{2}i^{2}})$$

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

$$Re(\frac{\left(1+5i\right)\left(7-6i\right)}{85})$$

Multiply complex numbers 1+5i and 7-6i like you multiply binomials.

$$Re(\frac{1\times 7+1\times \left(-6i\right)+5i\times 7+5\left(-6\right)i^{2}}{85})$$

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

$$Re(\frac{1\times 7+1\times \left(-6i\right)+5i\times 7+5\left(-6\right)\left(-1\right)}{85})$$

Do the multiplications in 1\times 7+1\times \left(-6i\right)+5i\times 7+5\left(-6\right)\left(-1\right).

$$Re(\frac{7-6i+35i+30}{85})$$

Combine the real and imaginary parts in 7-6i+35i+30.

$$Re(\frac{7+30+\left(-6+35\right)i}{85})$$

Do the additions in 7+30+\left(-6+35\right)i.

$$Re(\frac{37+29i}{85})$$

Divide 37+29i by 85 to get \frac{37}{85}+\frac{29}{85}i.

$$Re(\frac{37}{85}+\frac{29}{85}i)$$

The real part of \frac{37}{85}+\frac{29}{85}i is \frac{37}{85}.

$$\frac{37}{85}$$