To simplify the expression [tex]\(\frac{\sqrt{3} + 1}{2 \sqrt{2} - \sqrt{3}}\)[/tex], we will rationalize the denominator, which means we will eliminate the square root from the denominator.
### Step 1: Identifying the Conjugate
The conjugate of the denominator [tex]\(2 \sqrt{2} - \sqrt{3}\)[/tex] is [tex]\(2 \sqrt{2} + \sqrt{3}\)[/tex]. We will multiply both the numerator and the denominator by this conjugate to rationalize the denominator.
### Step 2: Multiply Numerator and Denominator by the Conjugate
[tex]\[
\frac{(\sqrt{3} + 1) (2 \sqrt{2} + \sqrt{3})}{(2 \sqrt{2} - \sqrt{3})(2 \sqrt{2} + \sqrt{3})}
\][/tex]
### Step 3: Simplify the Denominator using Difference of Squares
The denominator becomes:
[tex]\[
(2 \sqrt{2})^2 - (\sqrt{3})^2 = 4 \cdot 2 - 3 = 8 - 3 = 5
\][/tex]
### Step 4: Expand the Numerator
Now, expand [tex]\((\sqrt{3} + 1) (2 \sqrt{2} + \sqrt{3})\)[/tex]:
[tex]\[
(\sqrt{3} + 1) (2 \sqrt{2} + \sqrt{3}) = \sqrt{3} \cdot 2 \sqrt{2} + \sqrt{3} \cdot \sqrt{3} + 1 \cdot 2 \sqrt{2} + 1 \cdot \sqrt{3}
\][/tex]
[tex]\[
= 2 \sqrt{6} + 3 + 2 \sqrt{2} + \sqrt{3}
\][/tex]
### Step 5: Combine Like Terms in the Numerator
The simplified numerator is:
[tex]\[
2 \sqrt{6} + 3 + 2 \sqrt{2} + \sqrt{3}
\][/tex]
### Step 6: Divide by the Simplified Denominator
The final expression, after rationalizing the denominator, is:
[tex]\[
\frac{2 \sqrt{6} + 3 + 2 \sqrt{2} + \sqrt{3}}{5}
\][/tex]
### Step 7: Evaluate the Result
Numerically evaluating the expression, we get:
[tex]\[
2.4918914835762846
\][/tex]
Thus, the simplified form of the given expression [tex]\(\frac{\sqrt{3} + 1}{2 \sqrt{2} - \sqrt{3}}\)[/tex] is approximately [tex]\(2.4919\)[/tex].
