Given `sqrt(2)=1.4142`, find correct to three places of decimal the value of `(4)/(3sqrt(3)-2sqrt(2))+(3)/(3sqrt(3)+2sqrt(2))`

Options

A. `2.063`
B. `2.036`
C. `2.306`
D. `2.36`

Answer

To solve the expression \(\frac{4}{3\sqrt{3} - 2\sqrt{2}} + \frac{3}{3\sqrt{3} + 2\sqrt{2}}\) given that \(\sqrt{2} = 1.4142\), we will follow these steps:

### Step 1: Find a common denominator
The common denominator for the two fractions is \((3\sqrt{3} - 2\sqrt{2})(3\sqrt{3} + 2\sqrt{2})\).

### Step 2: Rewrite the expression
We can rewrite the expression as:
\[ \frac{4(3\sqrt{3} + 2\sqrt{2}) + 3(3\sqrt{3} - 2\sqrt{2})}{(3\sqrt{3} - 2\sqrt{2})(3\sqrt{3} + 2\sqrt{2})} \]

### Step 3: Expand the numerator
Now, we expand the numerator:
\[ 4(3\sqrt{3}) + 4(2\sqrt{2}) + 3(3\sqrt{3}) - 3(2\sqrt{2}) = 12\sqrt{3} + 8\sqrt{2} + 9\sqrt{3} - 6\sqrt{2} \]
Combining like terms gives:
\[ (12\sqrt{3} + 9\sqrt{3}) + (8\sqrt{2} - 6\sqrt{2}) = 21\sqrt{3} + 2\sqrt{2} \]

### Step 4: Simplify the denominator
The denominator simplifies using the difference of squares:
\[ (3\sqrt{3})^2 - (2\sqrt{2})^2 = 27 - 8 = 19 \]

### Step 5: Combine the results
Now we have:
\[ \frac{21\sqrt{3} + 2\sqrt{2}}{19} \]

### Step 6: Substitute the values of \(\sqrt{2}\) and \(\sqrt{3}\)
We know \(\sqrt{2} = 1.4142\) and we can approximate \(\sqrt{3} \approx 1.732\). Thus:
\[ 21\sqrt{3} = 21 \times 1.732 = 36.372 \]
\[ 2\sqrt{2} = 2 \times 1.4142 = 2.8284 \]

### Step 7: Add the results
Now we add these results:
\[ 36.372 + 2.8284 = 39.2004 \]

### Step 8: Divide by 19
Now we divide by 19:
\[ \frac{39.2004}{19} \approx 2.063 \]

### Final Answer
Thus, the value of the expression correct to three decimal places is:
\[ \boxed{2.063} \]