If `m=1+sqrt(2)`, then find the value of `m^(4)-(1)/(m^(4))` .

Answer

To solve the problem, we need to find the value of \( m^4 - \frac{1}{m^4} \) given that \( m = 1 + \sqrt{2} \).

### Step-by-Step Solution:

1. **Calculate \( m^2 \)**:
\[ m^2 = (1 + \sqrt{2})^2 = 1^2 + 2 \cdot 1 \cdot \sqrt{2} + (\sqrt{2})^2 = 1 + 2\sqrt{2} + 2 = 3 + 2\sqrt{2} \]

**Hint**: Use the formula \( (a + b)^2 = a^2 + 2ab + b^2 \).

2. **Calculate \( \frac{1}{m} \)**:
\[ \frac{1}{m} = \frac{1}{1 + \sqrt{2}} \cdot \frac{1 - \sqrt{2}}{1 - \sqrt{2}} = \frac{1 - \sqrt{2}}{(1 + \sqrt{2})(1 - \sqrt{2})} = \frac{1 - \sqrt{2}}{1 - 2} = \sqrt{2} - 1 \]

**Hint**: Rationalize the denominator by multiplying by the conjugate.

3. **Calculate \( \frac{1}{m^2} \)**:
\[ \frac{1}{m^2} = \left(\frac{1}{m}\right)^2 = (\sqrt{2} - 1)^2 = 2 - 2\sqrt{2} + 1 = 3 - 2\sqrt{2} \]

**Hint**: Again use the square formula for \( (a - b)^2 \).

4. **Calculate \( m^2 + \frac{1}{m^2} \)**:
\[ m^2 + \frac{1}{m^2} = (3 + 2\sqrt{2}) + (3 - 2\sqrt{2}) = 6 \]

**Hint**: Combine like terms.

5. **Calculate \( m^2 - \frac{1}{m^2} \)**:
\[ m^2 - \frac{1}{m^2} = (3 + 2\sqrt{2}) - (3 - 2\sqrt{2}) = 4\sqrt{2} \]

**Hint**: Again, combine like terms, focusing on the square root components.

6. **Use the identity for \( m^4 - \frac{1}{m^4} \)**:
\[ m^4 - \frac{1}{m^4} = (m^2 - \frac{1}{m^2})^2 + 2 \]
\[ m^4 - \frac{1}{m^4} = (4\sqrt{2})^2 + 2 = 32 + 2 = 34 \]

**Hint**: Remember the identity \( a^2 - b^2 = (a - b)(a + b) \) and how to square a binomial.

### Final Answer:
\[ m^4 - \frac{1}{m^4} = 34 \]