Solve 3+7i/7+6i

$$\frac { 3 + 7 i } { 7 + 6 i }$$

Evaluate

$$\frac{63}{85}+\frac{31}{85}i\approx 0.741176471+0.364705882i$$

Solution Steps

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

$$\frac{\left(3+7i\right)\left(7-6i\right)}{\left(7+6i\right)\left(7-6i\right)}$$

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

$$\frac{\left(3+7i\right)\left(7-6i\right)}{7^{2}-6^{2}i^{2}}$$

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

$$\frac{\left(3+7i\right)\left(7-6i\right)}{85}$$

Multiply complex numbers 3+7i and 7-6i like you multiply binomials.

$$\frac{3\times 7+3\times \left(-6i\right)+7i\times 7+7\left(-6\right)i^{2}}{85}$$

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

$$\frac{3\times 7+3\times \left(-6i\right)+7i\times 7+7\left(-6\right)\left(-1\right)}{85}$$

Do the multiplications in 3\times 7+3\times \left(-6i\right)+7i\times 7+7\left(-6\right)\left(-1\right).

$$\frac{21-18i+49i+42}{85}$$

Combine the real and imaginary parts in 21-18i+49i+42.

$$\frac{21+42+\left(-18+49\right)i}{85}$$

Do the additions in 21+42+\left(-18+49\right)i.

$$\frac{63+31i}{85}$$

Divide 63+31i by 85 to get \frac{63}{85}+\frac{31}{85}i.

$$\frac{63}{85}+\frac{31}{85}i$$

Real Part

$$\frac{63}{85} = 0.7411764705882353$$

Solution Steps

Multiply both numerator and denominator of \frac{3+7i}{7+6i} by the complex conjugate of the denominator, 7-6i.

$$Re(\frac{\left(3+7i\right)\left(7-6i\right)}{\left(7+6i\right)\left(7-6i\right)})$$

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

$$Re(\frac{\left(3+7i\right)\left(7-6i\right)}{7^{2}-6^{2}i^{2}})$$

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

$$Re(\frac{\left(3+7i\right)\left(7-6i\right)}{85})$$

Multiply complex numbers 3+7i and 7-6i like you multiply binomials.

$$Re(\frac{3\times 7+3\times \left(-6i\right)+7i\times 7+7\left(-6\right)i^{2}}{85})$$

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

$$Re(\frac{3\times 7+3\times \left(-6i\right)+7i\times 7+7\left(-6\right)\left(-1\right)}{85})$$

Do the multiplications in 3\times 7+3\times \left(-6i\right)+7i\times 7+7\left(-6\right)\left(-1\right).

$$Re(\frac{21-18i+49i+42}{85})$$

Combine the real and imaginary parts in 21-18i+49i+42.

$$Re(\frac{21+42+\left(-18+49\right)i}{85})$$

Do the additions in 21+42+\left(-18+49\right)i.

$$Re(\frac{63+31i}{85})$$

Divide 63+31i by 85 to get \frac{63}{85}+\frac{31}{85}i.

$$Re(\frac{63}{85}+\frac{31}{85}i)$$

The real part of \frac{63}{85}+\frac{31}{85}i is \frac{63}{85}.

$$\frac{63}{85}$$