Simplify sqrt(-5)(7 + sqrt(-8))

Problem

Simplify:

$$\sqrt{-5}\left(7+\sqrt{-8}\right)$$

Step 1: Rewrite Negative Square Roots

Since $\sqrt{-1}=i$, rewrite each square root of a negative number using the imaginary unit:

$$\sqrt{-5}=i\sqrt{5}$$

and

$$\sqrt{-8}=i\sqrt{8}.$$

So the expression becomes

$$i\sqrt{5}\left(7+i\sqrt{8}\right).$$

Step 2: Simplify the Radical Parts

Simplify the remaining positive radical:

$$\sqrt{8}=\sqrt{4\cdot 2}=2\sqrt{2}.$$

So the expression is now

$$i\sqrt{5}\left(7+2i\sqrt{2}\right).$$

Step 3: Distribute the Outside Factor

Distribute $i\sqrt{5}$ across the parentheses:

$$i\sqrt{5}\left(7+2i\sqrt{2}\right) =7i\sqrt{5}+2i^2\sqrt{10}.$$

Step 4: Use the Value of $i^2$

Since

$$i^2=-1,$$

the second term becomes

$$2i^2\sqrt{10}=-2\sqrt{10}.$$

So the expression is

$$7i\sqrt{5}-2\sqrt{10}.$$

Step 5: Write the Complex Result

Write the result in standard complex-number form, with the real part first and the imaginary part second:

$$-2\sqrt{10}+7i\sqrt{5}.$$

Thus, the simplified expression is

$$\boxed{-2\sqrt{10}+7i\sqrt{5}}.$$