The value of `(7+3sqrt""5)/(3+sqrt""5)-(7-3sqrt""5)/(3-sqrt""5)` lies between:

Options

A. 1.5 and 2
B. 3 and 3.5
C. 2.5 and 3
D. 2 and 2.5

Answer

To solve the expression \((7 + 3\sqrt{5})/(3 + \sqrt{5}) - (7 - 3\sqrt{5})/(3 - \sqrt{5})\), we will follow these steps:

### Step 1: Simplify Each Fraction
We start by simplifying each fraction separately.

For the first fraction:
\[ \frac{7 + 3\sqrt{5}}{3 + \sqrt{5}} \]
We can multiply the numerator and denominator by the conjugate of the denominator, which is \(3 - \sqrt{5}\):
\[ = \frac{(7 + 3\sqrt{5})(3 - \sqrt{5})}{(3 + \sqrt{5})(3 - \sqrt{5})} \]

For the second fraction:
\[ \frac{7 - 3\sqrt{5}}{3 - \sqrt{5}} \]
Similarly, we multiply the numerator and denominator by the conjugate of the denominator, which is \(3 + \sqrt{5}\):
\[ = \frac{(7 - 3\sqrt{5})(3 + \sqrt{5})}{(3 - \sqrt{5})(3 + \sqrt{5})} \]

### Step 2: Calculate the Denominators
The denominators for both fractions can be simplified using the difference of squares:
\[ (3 + \sqrt{5})(3 - \sqrt{5}) = 3^2 - (\sqrt{5})^2 = 9 - 5 = 4 \]

### Step 3: Expand the Numerators
Now we expand the numerators:

For the first fraction:
\[ (7 + 3\sqrt{5})(3 - \sqrt{5}) = 21 - 7\sqrt{5} + 9\sqrt{5} - 15 = 6 + 2\sqrt{5} \]

For the second fraction:
\[ (7 - 3\sqrt{5})(3 + \sqrt{5}) = 21 + 7\sqrt{5} - 9\sqrt{5} - 15 = 6 - 2\sqrt{5} \]

### Step 4: Combine the Fractions
Now we can combine the two fractions:
\[ \frac{6 + 2\sqrt{5}}{4} - \frac{6 - 2\sqrt{5}}{4} = \frac{(6 + 2\sqrt{5}) - (6 - 2\sqrt{5})}{4} \]
This simplifies to:
\[ \frac{6 + 2\sqrt{5} - 6 + 2\sqrt{5}}{4} = \frac{4\sqrt{5}}{4} = \sqrt{5} \]

### Step 5: Determine the Range of \(\sqrt{5}\)
We know that:
\[ \sqrt{5} \approx 2.236 \]
Thus, \(\sqrt{5}\) lies between 2 and 2.5.

### Final Answer
The value of the expression lies between \(2\) and \(2.5\).