Solve frac{sqrt{2}+isqrt{2}}{1+isqrt{3}}

$$\frac { \sqrt { 2 } + i \sqrt { 2 } } { 1 + i \sqrt { 3 } }$$

Evaluate

$$\sqrt{2}\left(\frac{1}{4}+\frac{1}{4}i\right)+\sqrt{6}\left(\frac{1}{4}-\frac{1}{4}i\right)\approx 0.965925826-0.258819045i$$

Solution Steps

Combine \sqrt{2} and i\sqrt{2} to get \left(1+i\right)\sqrt{2}.

$$\frac{\left(1+i\right)\sqrt{2}}{1+i\sqrt{3}}$$

Rationalize the denominator of \frac{\left(1+i\right)\sqrt{2}}{1+i\sqrt{3}} by multiplying numerator and denominator by 1-i\sqrt{3}.

$$\frac{\left(1+i\right)\sqrt{2}\left(1-i\sqrt{3}\right)}{\left(1+i\sqrt{3}\right)\left(1-i\sqrt{3}\right)}$$

Consider \left(1+i\sqrt{3}\right)\left(1-i\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(1+i\right)\sqrt{2}\left(1-i\sqrt{3}\right)}{1^{2}-\left(i\sqrt{3}\right)^{2}}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{\left(1+i\right)\sqrt{2}\left(1-i\sqrt{3}\right)}{1-\left(i\sqrt{3}\right)^{2}}$$

Expand \left(i\sqrt{3}\right)^{2}.

$$\frac{\left(1+i\right)\sqrt{2}\left(1-i\sqrt{3}\right)}{1-i^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate i to the power of 2 and get -1.

$$\frac{\left(1+i\right)\sqrt{2}\left(1-i\sqrt{3}\right)}{1-\left(-\left(\sqrt{3}\right)^{2}\right)}$$

The square of \sqrt{3} is 3.

$$\frac{\left(1+i\right)\sqrt{2}\left(1-i\sqrt{3}\right)}{1-\left(-3\right)}$$

Multiply -1 and -3 to get 3.

$$\frac{\left(1+i\right)\sqrt{2}\left(1-i\sqrt{3}\right)}{1+3}$$

Add 1 and 3 to get 4.

$$\frac{\left(1+i\right)\sqrt{2}\left(1-i\sqrt{3}\right)}{4}$$

Divide \left(1+i\right)\sqrt{2}\left(1-i\sqrt{3}\right) by 4 to get \left(\frac{1}{4}+\frac{1}{4}i\right)\sqrt{2}\left(1-i\sqrt{3}\right).

$$\left(\frac{1}{4}+\frac{1}{4}i\right)\sqrt{2}\left(1-i\sqrt{3}\right)$$

Use the distributive property to multiply \left(\frac{1}{4}+\frac{1}{4}i\right)\sqrt{2} by 1-i\sqrt{3}.

$$\left(\frac{1}{4}+\frac{1}{4}i\right)\sqrt{2}+\left(\frac{1}{4}-\frac{1}{4}i\right)\sqrt{3}\sqrt{2}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$\left(\frac{1}{4}+\frac{1}{4}i\right)\sqrt{2}+\left(\frac{1}{4}-\frac{1}{4}i\right)\sqrt{6}$$

Real Part

$$\frac{\sqrt{2} {(\sqrt{3} + 1)}}{4} = 0.9659258262890683$$