Solve 56+8i/14+10i

$$\frac { 56 + 8 i } { 14 + 10 i }$$

Evaluate

$$\frac{108}{37}-\frac{56}{37}i\approx 2.918918919-1.513513514i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 14-10i.

$$\frac{\left(56+8i\right)\left(14-10i\right)}{\left(14+10i\right)\left(14-10i\right)}$$

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

$$\frac{\left(56+8i\right)\left(14-10i\right)}{14^{2}-10^{2}i^{2}}$$

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

$$\frac{\left(56+8i\right)\left(14-10i\right)}{296}$$

Multiply complex numbers 56+8i and 14-10i like you multiply binomials.

$$\frac{56\times 14+56\times \left(-10i\right)+8i\times 14+8\left(-10\right)i^{2}}{296}$$

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

$$\frac{56\times 14+56\times \left(-10i\right)+8i\times 14+8\left(-10\right)\left(-1\right)}{296}$$

Do the multiplications in 56\times 14+56\times \left(-10i\right)+8i\times 14+8\left(-10\right)\left(-1\right).

$$\frac{784-560i+112i+80}{296}$$

Combine the real and imaginary parts in 784-560i+112i+80.

$$\frac{784+80+\left(-560+112\right)i}{296}$$

Do the additions in 784+80+\left(-560+112\right)i.

$$\frac{864-448i}{296}$$

Divide 864-448i by 296 to get \frac{108}{37}-\frac{56}{37}i.

$$\frac{108}{37}-\frac{56}{37}i$$

Real Part

$$\frac{108}{37} = 2\frac{34}{37} = 2.918918918918919$$

Solution Steps

Multiply both numerator and denominator of \frac{56+8i}{14+10i} by the complex conjugate of the denominator, 14-10i.

$$Re(\frac{\left(56+8i\right)\left(14-10i\right)}{\left(14+10i\right)\left(14-10i\right)})$$

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

$$Re(\frac{\left(56+8i\right)\left(14-10i\right)}{14^{2}-10^{2}i^{2}})$$

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

$$Re(\frac{\left(56+8i\right)\left(14-10i\right)}{296})$$

Multiply complex numbers 56+8i and 14-10i like you multiply binomials.

$$Re(\frac{56\times 14+56\times \left(-10i\right)+8i\times 14+8\left(-10\right)i^{2}}{296})$$

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

$$Re(\frac{56\times 14+56\times \left(-10i\right)+8i\times 14+8\left(-10\right)\left(-1\right)}{296})$$

Do the multiplications in 56\times 14+56\times \left(-10i\right)+8i\times 14+8\left(-10\right)\left(-1\right).

$$Re(\frac{784-560i+112i+80}{296})$$

Combine the real and imaginary parts in 784-560i+112i+80.

$$Re(\frac{784+80+\left(-560+112\right)i}{296})$$

Do the additions in 784+80+\left(-560+112\right)i.

$$Re(\frac{864-448i}{296})$$

Divide 864-448i by 296 to get \frac{108}{37}-\frac{56}{37}i.

$$Re(\frac{108}{37}-\frac{56}{37}i)$$

The real part of \frac{108}{37}-\frac{56}{37}i is \frac{108}{37}.

$$\frac{108}{37}$$