Solve (1+sqrt{3}i)(1+i)(sqrt{5}-i)

$$( 1 + \sqrt { 3 } i ) ( 1 + i ) ( \sqrt { 5 } - i )$$

Evaluate

$$\left(1+i\right)\left(\sqrt{5}-i\right)\left(1+\sqrt{3}i\right)\approx 1.095135439+6.841102131i$$

Solution Steps

Use the distributive property to multiply 1+i\sqrt{3} by 1+i.

$$\left(1+i+\left(-1+i\right)\sqrt{3}\right)\left(\sqrt{5}-i\right)$$

Apply the distributive property by multiplying each term of 1+i+\left(-1+i\right)\sqrt{3} by each term of \sqrt{5}-i.

$$\left(1+i\right)\sqrt{5}+\left(1-i\right)+\left(-1+i\right)\sqrt{3}\sqrt{5}+\left(1+i\right)\sqrt{3}$$

To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.

$$\left(1+i\right)\sqrt{5}+\left(1-i\right)+\left(-1+i\right)\sqrt{15}+\left(1+i\right)\sqrt{3}$$

Real Part

$$\sqrt{3} + \sqrt{5} + 1 - \sqrt{15} = 1.095135439$$