Evaluate `(sqrt2+1)^(5))-(sqrt2-1)^(5)` using binomial theorem.

Answer

To evaluate \((\sqrt{2}+1)^{5}-(\sqrt{2}-1)^{5}\) using the Binomial Theorem, we can follow these steps:

### Step 1: Apply the Binomial Theorem
The Binomial Theorem states that:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
In our case, we will expand \((\sqrt{2}+1)^{5}\) and \((\sqrt{2}-1)^{5}\).

### Step 2: Expand \((\sqrt{2}+1)^{5}\)
Using the Binomial Theorem:
\[ (\sqrt{2}+1)^{5} = \sum_{k=0}^{5} \binom{5}{k} (\sqrt{2})^{5-k} (1)^{k} \]
This gives us:
\[ = \binom{5}{0} (\sqrt{2})^{5} + \binom{5}{1} (\sqrt{2})^{4} (1) + \binom{5}{2} (\sqrt{2})^{3} (1)^{2} + \binom{5}{3} (\sqrt{2})^{2} (1)^{3} + \binom{5}{4} (\sqrt{2})^{1} (1)^{4} + \binom{5}{5} (1)^{5} \]

Calculating each term:
\[ = 1 \cdot (\sqrt{2})^{5} + 5 \cdot (\sqrt{2})^{4} + 10 \cdot (\sqrt{2})^{3} + 10 \cdot (\sqrt{2})^{2} + 5 \cdot (\sqrt{2})^{1} + 1 \]

### Step 3: Expand \((\sqrt{2}-1)^{5}\)
Similarly, we expand \((\sqrt{2}-1)^{5}\):
\[ (\sqrt{2}-1)^{5} = \sum_{k=0}^{5} \binom{5}{k} (\sqrt{2})^{5-k} (-1)^{k} \]
This gives us:
\[ = \binom{5}{0} (\sqrt{2})^{5} (-1)^{0} + \binom{5}{1} (\sqrt{2})^{4} (-1)^{1} + \binom{5}{2} (\sqrt{2})^{3} (-1)^{2} + \binom{5}{3} (\sqrt{2})^{2} (-1)^{3} + \binom{5}{4} (\sqrt{2})^{1} (-1)^{4} + \binom{5}{5} (-1)^{5} \]

Calculating each term:
\[ = 1 \cdot (\sqrt{2})^{5} - 5 \cdot (\sqrt{2})^{4} + 10 \cdot (\sqrt{2})^{3} - 10 \cdot (\sqrt{2})^{2} + 5 \cdot (\sqrt{2})^{1} - 1 \]

### Step 4: Subtract the two expansions
Now we subtract the two expansions:
\[ (\sqrt{2}+1)^{5} - (\sqrt{2}-1)^{5} \]
Combining like terms:
\[ = \left(1 \cdot (\sqrt{2})^{5} - 1 \cdot (\sqrt{2})^{5}\right) + \left(5 \cdot (\sqrt{2})^{4} + 5 \cdot (\sqrt{2})^{4}\right) + \left(10 \cdot (\sqrt{2})^{3} - 10 \cdot (\sqrt{2})^{3}\right) + \left(10 \cdot (\sqrt{2})^{2} + 10 \cdot (\sqrt{2})^{2}\right) + \left(5 \cdot (\sqrt{2})^{1} - 5 \cdot (\sqrt{2})^{1}\right) + \left(1 - (-1)\right) \]

This simplifies to:
\[ = 0 + 10 \cdot (\sqrt{2})^{4} + 0 + 20 \cdot (\sqrt{2})^{2} + 0 + 2 \]
Thus, we have:
\[ = 10 \cdot (\sqrt{2})^{4} + 20 \cdot (\sqrt{2})^{2} + 2 \]

### Step 5: Calculate the final values
Now we substitute \((\sqrt{2})^{2} = 2\) and \((\sqrt{2})^{4} = 4\):
\[ = 10 \cdot 4 + 20 \cdot 2 + 2 = 40 + 40 + 2 = 82 \]

### Final Answer
Thus, the value of \((\sqrt{2}+1)^{5}-(\sqrt{2}-1)^{5}\) is:
\[ \boxed{82} \]