Solve (sqrt{2}-3sqrt{5}i)(sqrt{6}-sqrt{8}i)

$$( \sqrt { 2 } - 3 \sqrt { 5 } i ) ( \sqrt { 6 } - \sqrt { 8 } i )$$

Evaluate

$$-3\sqrt{30}i+2\sqrt{3}-6\sqrt{10}-4i\approx -15.509564346-20.431676725i$$

Solution Steps

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\left(\sqrt{2}-3i\sqrt{5}\right)\left(\sqrt{6}-i\times 2\sqrt{2}\right)$$

Multiply -1 and 3i to get -3i.

$$\left(\sqrt{2}-3i\sqrt{5}\right)\left(\sqrt{6}-2i\sqrt{2}\right)$$

Apply the distributive property by multiplying each term of \sqrt{2}-3i\sqrt{5} by each term of \sqrt{6}-2i\sqrt{2}.

$$\sqrt{2}\sqrt{6}-2i\left(\sqrt{2}\right)^{2}-3i\sqrt{5}\sqrt{6}-6\sqrt{5}\sqrt{2}$$

Factor 6=2\times 3. Rewrite the square root of the product \sqrt{2\times 3} as the product of square roots \sqrt{2}\sqrt{3}.

$$\sqrt{2}\sqrt{2}\sqrt{3}-2i\left(\sqrt{2}\right)^{2}-3i\sqrt{5}\sqrt{6}-6\sqrt{5}\sqrt{2}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$2\sqrt{3}-2i\left(\sqrt{2}\right)^{2}-3i\sqrt{5}\sqrt{6}-6\sqrt{5}\sqrt{2}$$

The square of \sqrt{2} is 2.

$$2\sqrt{3}-2i\times 2-3i\sqrt{5}\sqrt{6}-6\sqrt{5}\sqrt{2}$$

Multiply -2i and 2 to get -4i.

$$2\sqrt{3}-4i-3i\sqrt{5}\sqrt{6}-6\sqrt{5}\sqrt{2}$$

To multiply \sqrt{5} and \sqrt{6}, multiply the numbers under the square root.

$$2\sqrt{3}-4i-3i\sqrt{30}-6\sqrt{5}\sqrt{2}$$

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

$$2\sqrt{3}-4i-3i\sqrt{30}-6\sqrt{10}$$

Real Part

$$2 {(\sqrt{3} - 3 \sqrt{10})} = -15.509564346$$