Solve ysqrt{-20}(sqrt{5}-sqrt{-5})

$$y \sqrt { - 20 } ( \sqrt { 5 } - \sqrt { - 5 } )$$

Evaluate (complex solution)

$$\left(10+10i\right)y$$

Solution Steps

Factor -20=\left(2i\right)^{2}\times 5. Rewrite the square root of the product \sqrt{\left(2i\right)^{2}\times 5} as the product of square roots \sqrt{\left(2i\right)^{2}}\sqrt{5}. Take the square root of \left(2i\right)^{2}.

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

Factor -5=5\left(-1\right). Rewrite the square root of the product \sqrt{5\left(-1\right)} as the product of square roots \sqrt{5}\sqrt{-1}. By definition, the square root of -1 is i.

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

Multiply -1 and i to get -i.

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

Combine \sqrt{5} and -i\sqrt{5} to get \left(1-i\right)\sqrt{5}.

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

Multiply 2i times 1-i.

$$y\left(2i\times 1+2\left(-1\right)i^{2}\right)\sqrt{5}\sqrt{5}$$

By definition, i^{2} is -1.

$$y\left(2i\times 1+2\left(-1\right)\left(-1\right)\right)\sqrt{5}\sqrt{5}$$

Do the multiplications. Reorder the terms.

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

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$y\left(2+2i\right)\times 5$$

Multiply 2+2i times 5.

$$y\left(2\times 5+2i\times 5\right)$$

Do the multiplications.

$$y\left(10+10i\right)$$

Evaluate

$$\text{Indeterminate}$$