What is the following quotient?

[tex]\[ \frac{2-\sqrt{8}}{4+\sqrt{12}} \][/tex]

A. [tex]\(\frac{\sqrt{3}-\sqrt{6}}{4}\)[/tex]

B. [tex]\(\frac{2+\sqrt{3}-2\sqrt{2}-\sqrt{6}}{4}\)[/tex]

C. [tex]\(2-\sqrt{3}-2\sqrt{2}+\sqrt{6}\)[/tex]

D. [tex]\(\frac{-2-\sqrt{3}+2\sqrt{2}+\sqrt{6}}{2}\)[/tex]



Answer :

To solve the quotient

[tex]\[ \frac{2 - \sqrt{8}}{4 + \sqrt{12}} \][/tex]

we need to simplify both the numerator and the denominator.

### Step 1: Simplify the numerator and denominator

First, let's look at the numerator:

[tex]\[ 2 - \sqrt{8} \][/tex]

We know that [tex]\( \sqrt{8} = 2\sqrt{2} \)[/tex], so we can rewrite the numerator as:

[tex]\[ 2 - 2\sqrt{2} \][/tex]

Next, the denominator:

[tex]\[ 4 + \sqrt{12} \][/tex]

We know that [tex]\( \sqrt{12} = 2\sqrt{3} \)[/tex], so we can rewrite the denominator as:

[tex]\[ 4 + 2\sqrt{3} \][/tex]

### Step 2: Rationalize the denominator

To get rid of the surd in the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\( 4 + 2\sqrt{3} \)[/tex] is [tex]\( 4 - 2\sqrt{3} \)[/tex].

So we multiply:

[tex]\[ \frac{(2 - 2\sqrt{2})(4 - 2\sqrt{3})}{(4 + 2\sqrt{3})(4 - 2\sqrt{3})} \][/tex]

### Step 3: Expand the numerator and denominator

Let's expand the numerator [tex]\((2 - 2\sqrt{2})(4 - 2\sqrt{3})\)[/tex]:

[tex]\[ \begin{align*} (2 - 2\sqrt{2})(4 - 2\sqrt{3}) &= 2 \cdot 4 - 2 \cdot 2\sqrt{3} - 2\sqrt{2} \cdot 4 + 2\sqrt{2} \cdot 2\sqrt{3} \\ &= 8 - 4\sqrt{3} - 8\sqrt{2} + 4\sqrt{6} \end{align*} \][/tex]

Now, expand the denominator [tex]\((4 + 2\sqrt{3})(4 - 2\sqrt{3})\)[/tex]:

[tex]\[ \begin{align*} (4 + 2\sqrt{3})(4 - 2\sqrt{3}) &= 4 \cdot 4 - (2\sqrt{3})^2 \\ &= 16 - 12 \\ &= 4 \end{align*} \][/tex]

### Step 4: Simplify the final expression

Putting it all together:

[tex]\[ \frac{8 - 4\sqrt{3} - 8\sqrt{2} + 4\sqrt{6}}{4} \][/tex]

We can simplify this by dividing each term in the numerator by 4:

[tex]\[ 8/4 - 4\sqrt{3}/4 - 8\sqrt{2}/4 + 4\sqrt{6}/4 = 2 - \sqrt{3} - 2\sqrt{2} + \sqrt{6} \][/tex]

### Result

Thus, the final simplified form of the quotient is:

[tex]\[ 2 - \sqrt{3} - 2\sqrt{2} + \sqrt{6} \][/tex]

This is the answer to the question.