Given that: `sqrt(3) = 1.732, (sqrt(6) + sqrt(2)) / (sqrt(6)-sqrt(2))` is equal to:

Options

A. 3.713
B. 3.721
C. 3.732
D. 3.752

Answer

To solve the expression \((\sqrt{6} + \sqrt{2}) / (\sqrt{6} - \sqrt{2})\), we can rationalize the denominator. Here’s a step-by-step solution:

### Step 1: Write down the expression
We start with the expression:
\[ \frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}} \]

### Step 2: Multiply by the conjugate
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is \((\sqrt{6} + \sqrt{2})\):
\[ \frac{(\sqrt{6} + \sqrt{2})(\sqrt{6} + \sqrt{2})}{(\sqrt{6} - \sqrt{2})(\sqrt{6} + \sqrt{2})} \]

### Step 3: Simplify the numerator
Now, we simplify the numerator:
\[ (\sqrt{6} + \sqrt{2})^2 = (\sqrt{6})^2 + 2(\sqrt{6})(\sqrt{2}) + (\sqrt{2})^2 \]
Calculating each term:
- \((\sqrt{6})^2 = 6\)
- \((\sqrt{2})^2 = 2\)
- \(2(\sqrt{6})(\sqrt{2}) = 2\sqrt{12} = 4\sqrt{3}\)

So, the numerator becomes:
\[ 6 + 2 + 4\sqrt{3} = 8 + 4\sqrt{3} \]

### Step 4: Simplify the denominator
Now, we simplify the denominator:
\[ (\sqrt{6})^2 - (\sqrt{2})^2 = 6 - 2 = 4 \]

### Step 5: Combine the results
Now we can combine the results:
\[ \frac{8 + 4\sqrt{3}}{4} \]

### Step 6: Simplify the fraction
We can simplify this fraction by dividing each term in the numerator by 4:
\[ \frac{8}{4} + \frac{4\sqrt{3}}{4} = 2 + \sqrt{3} \]

### Final Result
Thus, the value of the expression \(\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\) is:
\[ 2 + \sqrt{3} \]

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