Solve frac{3+sqrt{2}i}{5i-sqrt{2}}

$$\frac{ 3+ \sqrt{ 2 } i }{ 5i- \sqrt{ 2 } }$$

Evaluate

$$\frac{3+\sqrt{2}i}{-\sqrt{2}+5i}\approx 0.10475656-0.62962963i$$

Solution Steps

Rationalize the denominator of \frac{3+i\sqrt{2}}{5i-\sqrt{2}} by multiplying numerator and denominator by 5i+\sqrt{2}.

$$\frac{\left(3+i\sqrt{2}\right)\left(5i+\sqrt{2}\right)}{\left(5i-\sqrt{2}\right)\left(5i+\sqrt{2}\right)}$$

Consider \left(5i-\sqrt{2}\right)\left(5i+\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(3+i\sqrt{2}\right)\left(5i+\sqrt{2}\right)}{\left(5i\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

Square 5i. Square \sqrt{2}.

$$\frac{\left(3+i\sqrt{2}\right)\left(5i+\sqrt{2}\right)}{-25-2}$$

Subtract 2 from -25 to get -27.

$$\frac{\left(3+i\sqrt{2}\right)\left(5i+\sqrt{2}\right)}{-27}$$

Apply the distributive property by multiplying each term of 3+i\sqrt{2} by each term of 5i+\sqrt{2}.

$$\frac{15i+3\sqrt{2}-5\sqrt{2}+i\left(\sqrt{2}\right)^{2}}{-27}$$

Combine 3\sqrt{2} and -5\sqrt{2} to get -2\sqrt{2}.

$$\frac{15i-2\sqrt{2}+i\left(\sqrt{2}\right)^{2}}{-27}$$

The square of \sqrt{2} is 2.

$$\frac{15i-2\sqrt{2}+2i}{-27}$$

Add 15i and 2i to get 17i.

$$\frac{17i-2\sqrt{2}}{-27}$$

Multiply both numerator and denominator by -1.

$$\frac{-17i+2\sqrt{2}}{27}$$

Real Part

$$-\sqrt{2}Im(\frac{1}{-\sqrt{2}+5i})+3Re(\frac{1}{-\sqrt{2}+5i})$$