Expand `(x+y)^(4)-(x-y)^(4)`. Hence find the value of `(3+sqrt(5))^(4) -(3-sqrt(5))^(4)`.

Explanation

To solve the problem, we need to expand the expression \((x+y)^{4} - (x-y)^{4}\) and then use this result to find the value of \((3+\sqrt{5})^{4} - (3-\sqrt{5})^{4}\).

### Step-by-Step Solution:

**Step 1: Expand \((x+y)^{4}\)**
Using the Binomial Theorem, we have:
\[ (x+y)^{4} = \sum_{k=0}^{4} \binom{4}{k} x^{4-k} y^{k} \]
This gives us:
\[ = \binom{4}{0} x^{4} + \binom{4}{1} x^{3}y + \binom{4}{2} x^{2}y^{2} + \binom{4}{3} xy^{3} + \binom{4}{4} y^{4} \]
Calculating the binomial coefficients:
\[ = 1 \cdot x^{4} + 4 \cdot x^{3}y + 6 \cdot x^{2}y^{2} + 4 \cdot xy^{3} + 1 \cdot y^{4} \]
Thus:
\[ (x+y)^{4} = x^{4} + 4x^{3}y + 6x^{2}y^{2} + 4xy^{3} + y^{4} \]

**Step 2: Expand \((x-y)^{4}\)**
Similarly, we can expand \((x-y)^{4}\):
\[ (x-y)^{4} = \sum_{k=0}^{4} \binom{4}{k} x^{4-k} (-y)^{k} \]
This gives us:
\[ = \binom{4}{0} x^{4} - \binom{4}{1} x^{3}y + \binom{4}{2} x^{2}y^{2} - \binom{4}{3} xy^{3} + \binom{4}{4} y^{4} \]
Calculating the binomial coefficients:
\[ = 1 \cdot x^{4} - 4 \cdot x^{3}y + 6 \cdot x^{2}y^{2} - 4 \cdot xy^{3} + 1 \cdot y^{4} \]
Thus:
\[ (x-y)^{4} = x^{4} - 4x^{3}y + 6x^{2}y^{2} - 4xy^{3} + y^{4} \]

**Step 3: Subtract the two expansions**
Now we compute:
\[ (x+y)^{4} - (x-y)^{4} = \left( x^{4} + 4x^{3}y + 6x^{2}y^{2} + 4xy^{3} + y^{4} \right) - \left( x^{4} - 4x^{3}y + 6x^{2}y^{2} - 4xy^{3} + y^{4} \right) \]
Simplifying this:
\[ = (x^{4} - x^{4}) + (4x^{3}y + 4x^{3}y) + (6x^{2}y^{2} - 6x^{2}y^{2}) + (4xy^{3} + 4xy^{3}) + (y^{4} - y^{4}) \]
This reduces to:
\[ = 8x^{3}y + 8xy^{3} \]
Factoring out \(8xy\):
\[ = 8xy(x^{2} + y^{2}) \]

**Step 4: Substitute \(x = 3\) and \(y = \sqrt{5}\)**
Now we need to find:
\[ (3+\sqrt{5})^{4} - (3-\sqrt{5})^{4} \]
Using our previous result:
\[ = 8xy(x^{2} + y^{2}) \]
Substituting \(x = 3\) and \(y = \sqrt{5}\):
\[ = 8 \cdot 3 \cdot \sqrt{5} \left(3^{2} + (\sqrt{5})^{2}\right) \]
Calculating \(3^{2} + (\sqrt{5})^{2}\):
\[ = 9 + 5 = 14 \]
Thus:
\[ = 8 \cdot 3 \cdot \sqrt{5} \cdot 14 \]
Calculating:
\[ = 336\sqrt{5} \]

### Final Answer:
\[ (3+\sqrt{5})^{4} - (3-\sqrt{5})^{4} = 336\sqrt{5} \]