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

Answer

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

### Step 1: Identify the expression
We start with the expression:
\[ \frac{2\sqrt{5} + 3\sqrt{2}}{2\sqrt{5} - 3\sqrt{2}} \]

### Step 2: Multiply by the conjugate
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(2\sqrt{5} - 3\sqrt{2}\) is \(2\sqrt{5} + 3\sqrt{2}\).

So, we multiply:
\[ \frac{2\sqrt{5} + 3\sqrt{2}}{2\sqrt{5} - 3\sqrt{2}} \cdot \frac{2\sqrt{5} + 3\sqrt{2}}{2\sqrt{5} + 3\sqrt{2}} \]

### Step 3: Apply the formula for the difference of squares
The denominator now becomes:
\[ (2\sqrt{5})^2 - (3\sqrt{2})^2 \]

Calculating this gives:
\[ = 4 \cdot 5 - 9 \cdot 2 = 20 - 18 = 2 \]

### Step 4: Expand the numerator
Now we expand the numerator:
\[ (2\sqrt{5} + 3\sqrt{2})^2 \]

Using the formula \((a + b)^2 = a^2 + 2ab + b^2\):
\[ = (2\sqrt{5})^2 + 2(2\sqrt{5})(3\sqrt{2}) + (3\sqrt{2})^2 \]
Calculating each term:
\[ = 4 \cdot 5 + 12\sqrt{10} + 9 \cdot 2 = 20 + 12\sqrt{10} + 18 = 38 + 12\sqrt{10} \]

### Step 5: Combine the results
Now we can write the entire expression:
\[ \frac{38 + 12\sqrt{10}}{2} \]

### Step 6: Simplify the fraction
We can simplify this by dividing each term in the numerator by 2:
\[ = \frac{38}{2} + \frac{12\sqrt{10}}{2} = 19 + 6\sqrt{10} \]

### Final Result
Thus, the rationalized form of the expression is:
\[ 19 + 6\sqrt{10} \]

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