Solve (sqrt{5}-sqrt{3}i)(sqrt{5}+sqrt{2}i)

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

Evaluate

$$-\sqrt{15}i+\sqrt{6}+5+\sqrt{10}i\approx 7.449489743-0.710705686i$$

Solution Steps

Use the distributive property to multiply \sqrt{5}-i\sqrt{3} by \sqrt{5}+i\sqrt{2}.

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

Use the distributive property to multiply \sqrt{5}-i\sqrt{3} by \sqrt{5}.

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

The square of \sqrt{5} is 5.

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

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

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

Use the distributive property to multiply i by \sqrt{5}-i\sqrt{3}.

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

Use the distributive property to multiply i\sqrt{5}+\sqrt{3} by \sqrt{2}.

$$5-i\sqrt{15}+i\sqrt{5}\sqrt{2}+\sqrt{3}\sqrt{2}$$

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$5-i\sqrt{15}+i\sqrt{10}+\sqrt{3}\sqrt{2}$$

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

$$5-i\sqrt{15}+i\sqrt{10}+\sqrt{6}$$

Real Part

$$\sqrt{6} + 5 = 7.449489743$$