Simplify :
`(22)/(2sqrt(3)+1)+(17)/(2sqrt(3)-1)`

Answer

To simplify the expression \(\frac{22}{2\sqrt{3}+1} + \frac{17}{2\sqrt{3}-1}\), we will follow these steps:

### Step 1: Identify the expression
We start with the given expression:
\[ \frac{22}{2\sqrt{3}+1} + \frac{17}{2\sqrt{3}-1} \]

### Step 2: Find the LCM
To combine the fractions, we need to find a common denominator. The LCM of the denominators \(2\sqrt{3}+1\) and \(2\sqrt{3}-1\) is:
\[ (2\sqrt{3}+1)(2\sqrt{3}-1) \]

### Step 3: Rewrite the fractions
Now we can rewrite each fraction with the common denominator:
\[ \frac{22(2\sqrt{3}-1)}{(2\sqrt{3}+1)(2\sqrt{3}-1)} + \frac{17(2\sqrt{3}+1)}{(2\sqrt{3}+1)(2\sqrt{3}-1)} \]

### Step 4: Combine the numerators
Now, we can combine the numerators:
\[ \frac{22(2\sqrt{3}-1) + 17(2\sqrt{3}+1)}{(2\sqrt{3}+1)(2\sqrt{3}-1)} \]

### Step 5: Expand the numerators
Expanding the numerators:
\[ 22(2\sqrt{3}-1) = 44\sqrt{3} - 22 \]
\[ 17(2\sqrt{3}+1) = 34\sqrt{3} + 17 \]
Now, combine these:
\[ 44\sqrt{3} - 22 + 34\sqrt{3} + 17 = (44\sqrt{3} + 34\sqrt{3}) + (-22 + 17) = 78\sqrt{3} - 5 \]

### Step 6: Simplify the denominator
Now, simplify the denominator using the difference of squares:
\[ (2\sqrt{3})^2 - 1^2 = 12 - 1 = 11 \]

### Step 7: Write the final expression
Putting it all together, we have:
\[ \frac{78\sqrt{3} - 5}{11} \]

### Final Answer
Thus, the simplified expression is:
\[ \frac{78\sqrt{3} - 5}{11} \]

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