Solve 53/7+2i

$$\frac { 53 } { 7 + 2 i }$$

Evaluate

$$7-2i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 7-2i.

$$\frac{53\left(7-2i\right)}{\left(7+2i\right)\left(7-2i\right)}$$

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

$$\frac{53\left(7-2i\right)}{7^{2}-2^{2}i^{2}}$$

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

$$\frac{53\left(7-2i\right)}{53}$$

Multiply 53 times 7-2i.

$$\frac{53\times 7+53\times \left(-2i\right)}{53}$$

Do the multiplications in 53\times 7+53\times \left(-2i\right).

$$\frac{371-106i}{53}$$

Divide 371-106i by 53 to get 7-2i.

$$7-2i$$

Real Part

$$7$$

Solution Steps

Multiply both numerator and denominator of \frac{53}{7+2i} by the complex conjugate of the denominator, 7-2i.

$$Re(\frac{53\left(7-2i\right)}{\left(7+2i\right)\left(7-2i\right)})$$

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

$$Re(\frac{53\left(7-2i\right)}{7^{2}-2^{2}i^{2}})$$

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

$$Re(\frac{53\left(7-2i\right)}{53})$$

Multiply 53 times 7-2i.

$$Re(\frac{53\times 7+53\times \left(-2i\right)}{53})$$

Do the multiplications in 53\times 7+53\times \left(-2i\right).

$$Re(\frac{371-106i}{53})$$

Divide 371-106i by 53 to get 7-2i.

$$Re(7-2i)$$

The real part of 7-2i is 7.

$$7$$