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

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

Evaluate

$$-\frac{4\sqrt{2}}{3}+\frac{7}{3}i\approx -1.885618083+2.333333333i$$

Solution Steps

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

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

Consider \left(i+\sqrt{2}\right)\left(i-\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}.

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

Square i. Square \sqrt{2}.

$$\frac{\left(i\sqrt{2}-5\right)\left(i-\sqrt{2}\right)}{-1-2}$$

Subtract 2 from -1 to get -3.

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

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

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

The square of \sqrt{2} is 2.

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

Multiply -i and 2 to get -2i.

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

Subtract 5i from -2i to get -7i.

$$\frac{-\sqrt{2}-7i+5\sqrt{2}}{-3}$$

Combine -\sqrt{2} and 5\sqrt{2} to get 4\sqrt{2}.

$$\frac{4\sqrt{2}-7i}{-3}$$

Multiply both numerator and denominator by -1.

$$\frac{-4\sqrt{2}+7i}{3}$$

Real Part

$$-\frac{4 \sqrt{2}}{3} = -1.885618083164127$$