`(2sqrt(6))/(sqrt(2)+sqrt(3)+sqrt(5))` equals

Options

A. `sqrt(2)+sqrt(3)-sqrt(5)`
B. `4-sqrt(2)-sqrt(3)`
C. `sqrt(2)+sqrt(3)+sqrt(6)-5`
D. `(1)/(2)(sqrt(2)+sqrt(5)-sqrt(3))`

Answer

To simplify the expression \(\frac{2\sqrt{6}}{\sqrt{2} + \sqrt{3} + \sqrt{5}}\), we can follow these steps:

### Step 1: Rationalize the Denominator
To rationalize the denominator, we will multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(\sqrt{2} + \sqrt{3} + \sqrt{5}\) can be expressed as \(\sqrt{2} + \sqrt{3} - \sqrt{5}\).

\[ \frac{2\sqrt{6}}{\sqrt{2} + \sqrt{3} + \sqrt{5}} \cdot \frac{\sqrt{2} + \sqrt{3} - \sqrt{5}}{\sqrt{2} + \sqrt{3} - \sqrt{5}} = \frac{2\sqrt{6}(\sqrt{2} + \sqrt{3} - \sqrt{5})}{(\sqrt{2} + \sqrt{3} + \sqrt{5})(\sqrt{2} + \sqrt{3} - \sqrt{5})} \]

### Step 2: Simplify the Denominator
Now, we will simplify the denominator using the difference of squares formula, \(a^2 - b^2\):

\[ (\sqrt{2} + \sqrt{3})^2 - (\sqrt{5})^2 \]

Calculating \((\sqrt{2} + \sqrt{3})^2\):

\[ (\sqrt{2})^2 + 2(\sqrt{2})(\sqrt{3}) + (\sqrt{3})^2 = 2 + 2\sqrt{6} + 3 = 5 + 2\sqrt{6} \]

Now, substituting back into the denominator:

\[ (5 + 2\sqrt{6}) - 5 = 2\sqrt{6} \]

### Step 3: Substitute Back into the Expression
Now we can substitute this back into our expression:

\[ \frac{2\sqrt{6}(\sqrt{2} + \sqrt{3} - \sqrt{5})}{2\sqrt{6}} \]

### Step 4: Cancel Out Common Terms
The \(2\sqrt{6}\) in the numerator and denominator cancels out:

\[ \sqrt{2} + \sqrt{3} - \sqrt{5} \]

### Final Answer
Thus, the simplified form of the expression is:

\[ \sqrt{2} + \sqrt{3} - \sqrt{5} \]

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