Question

multiply.

1. ((x - sqrt{8})(x + sqrt{8}))
2. (sqrt{12}(sqrt{3} + sqrt{6}))

Explanation:

Step1: Use difference of squares formula

Recall $(a-b)(a+b)=a^2-b^2$. Here $a=x$, $b=\sqrt{8}$.

$$\begin{align*} (x-\sqrt{8})(x+\sqrt{8})&=x^2-(\sqrt{8})^2 \end{align*}$$

Step2: Simplify the squared radical

$(\sqrt{8})^2=8$, so:
$x^2 - 8$

Step3: Distribute $\sqrt{12}$ to each term

Use distributive property $a(b+c)=ab+ac$. Here $a=\sqrt{12}$, $b=\sqrt{3}$, $c=\sqrt{6}$.

$$\begin{align*} \sqrt{12}(\sqrt{3}+\sqrt{6})&=\sqrt{12}\times\sqrt{3}+\sqrt{12}\times\sqrt{6} \end{align*}$$

Step4: Multiply radicals and simplify

Use $\sqrt{a}\times\sqrt{b}=\sqrt{ab}$:
$\sqrt{12\times3}=\sqrt{36}=6$, $\sqrt{12\times6}=\sqrt{72}=6\sqrt{2}$
$6 + 6\sqrt{2}$

Answer:

1. $x^2 - 8$
2. $6 + 6\sqrt{2}$