Question

simplify the following square root expression.
$(4 + \sqrt{-2})^2$
answer

Explanation:

Step1: Expand using square formula

Recall $(a+b)^2=a^2+2ab+b^2$. Let $a=4$, $b=\sqrt{-2}$.
$$(4+\sqrt{-2})^2=4^2+2\times4\times\sqrt{-2}+(\sqrt{-2})^2$$

Step2: Compute each term

Calculate individual components:
$4^2=16$, $2\times4\times\sqrt{-2}=8\sqrt{-2}$, $(\sqrt{-2})^2=-2$
$$=16+8\sqrt{-2}-2$$

Step3: Combine real terms

Simplify the real number part:
$$=14+8\sqrt{-2}$$

Step4: Rewrite with imaginary unit

Use $i=\sqrt{-1}$, so $\sqrt{-2}=i\sqrt{2}$:
$$=14+8i\sqrt{2}$$

Answer:

$14 + 8\sqrt{2}i$