Rationalize the denominatior of :
`(3)/(sqrt(5)+sqrt(2))`

Answer

To rationalize the denominator of the expression \(\frac{3}{\sqrt{5} + \sqrt{2}}\), we will follow these steps:

### Step 1: Identify the Conjugate
The first step in rationalizing the denominator is to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(\sqrt{5} + \sqrt{2}\) is \(\sqrt{5} - \sqrt{2}\).

### Step 2: Multiply by the Conjugate
Now, we multiply the numerator and the denominator by \(\sqrt{5} - \sqrt{2}\):

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

### Step 3: Simplify the Denominator
Next, we simplify the denominator using the difference of squares formula \(a^2 - b^2\):

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

### Step 4: Simplify the Numerator
Now, we simplify the numerator:

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

### Step 5: Combine the Results
Putting it all together, we have:

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

### Step 6: Cancel Common Factors
Now, we can cancel the common factor of 3 in the numerator and denominator:

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

### Final Answer
Thus, the rationalized form of \(\frac{3}{\sqrt{5} + \sqrt{2}}\) is:

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

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