Solve frac{2.8texttt{i}}{4+2.8texttt{i}}*10

$$\frac{ 2.8 \texttt{i} }{ 4+2.8 \texttt{i} } \times 10$$

Evaluate

$$\frac{490}{149}+\frac{700}{149}i\approx 3.288590604+4.697986577i$$

Solution Steps

Multiply both numerator and denominator of \frac{2.8i}{4+2.8i} by the complex conjugate of the denominator, 4-2.8i.

$$\frac{2.8i\left(4-2.8i\right)}{\left(4+2.8i\right)\left(4-2.8i\right)}\times 10$$

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

$$\frac{2.8i\left(4-2.8i\right)}{4^{2}-2.8^{2}i^{2}}\times 10$$

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

$$\frac{2.8i\left(4-2.8i\right)}{23.84}\times 10$$

Multiply 2.8i times 4-2.8i.

$$\frac{2.8i\times 4+2.8\left(-2.8\right)i^{2}}{23.84}\times 10$$

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

$$\frac{2.8i\times 4+2.8\left(-2.8\right)\left(-1\right)}{23.84}\times 10$$

Do the multiplications in 2.8i\times 4+2.8\left(-2.8\right)\left(-1\right). Reorder the terms.

$$\frac{7.84+11.2i}{23.84}\times 10$$

Divide 7.84+11.2i by 23.84 to get \frac{49}{149}+\frac{70}{149}i.

$$\left(\frac{49}{149}+\frac{70}{149}i\right)\times 10$$

Multiply \frac{49}{149}+\frac{70}{149}i times 10.

$$\frac{49}{149}\times 10+\frac{70}{149}i\times 10$$

Do the multiplications.

$$\frac{490}{149}+\frac{700}{149}i$$

Real Part

$$\frac{490}{149} = 3\frac{43}{149} = 3.2885906040268456$$

Solution Steps

Multiply both numerator and denominator of \frac{2.8i}{4+2.8i} by the complex conjugate of the denominator, 4-2.8i.

$$Re(\frac{2.8i\left(4-2.8i\right)}{\left(4+2.8i\right)\left(4-2.8i\right)}\times 10)$$

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

$$Re(\frac{2.8i\left(4-2.8i\right)}{4^{2}-2.8^{2}i^{2}}\times 10)$$

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

$$Re(\frac{2.8i\left(4-2.8i\right)}{23.84}\times 10)$$

Multiply 2.8i times 4-2.8i.

$$Re(\frac{2.8i\times 4+2.8\left(-2.8\right)i^{2}}{23.84}\times 10)$$

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

$$Re(\frac{2.8i\times 4+2.8\left(-2.8\right)\left(-1\right)}{23.84}\times 10)$$

Do the multiplications in 2.8i\times 4+2.8\left(-2.8\right)\left(-1\right). Reorder the terms.

$$Re(\frac{7.84+11.2i}{23.84}\times 10)$$

Divide 7.84+11.2i by 23.84 to get \frac{49}{149}+\frac{70}{149}i.

$$Re(\left(\frac{49}{149}+\frac{70}{149}i\right)\times 10)$$

Multiply \frac{49}{149}+\frac{70}{149}i times 10.

$$Re(\frac{49}{149}\times 10+\frac{70}{149}i\times 10)$$

Do the multiplications in \frac{49}{149}\times 10+\frac{70}{149}i\times 10.

$$Re(\frac{490}{149}+\frac{700}{149}i)$$

The real part of \frac{490}{149}+\frac{700}{149}i is \frac{490}{149}.

$$\frac{490}{149}$$