If ` m^(4) + (1)/(m^(4))` = 119 then ` m - (1)/(m) = ? `

Options

A. `pm 3`
B. 4
C. `pm 2`
D. `pm 1 `

Answer

To solve the equation \( m^4 + \frac{1}{m^4} = 119 \) and find \( m - \frac{1}{m} \), we can follow these steps:

### Step 1: Rewrite the given equation
We start with the equation:
\[ m^4 + \frac{1}{m^4} = 119 \]

### Step 2: Use the identity for \( m^2 + \frac{1}{m^2} \)
We can use the identity:
\[ m^4 + \frac{1}{m^4} = \left(m^2 + \frac{1}{m^2}\right)^2 - 2 \]
Let \( x = m^2 + \frac{1}{m^2} \). Then we can rewrite our equation as:
\[ x^2 - 2 = 119 \]

### Step 3: Solve for \( x^2 \)
Now, we can solve for \( x^2 \):
\[ x^2 = 119 + 2 = 121 \]

### Step 4: Solve for \( x \)
Taking the square root of both sides, we find:
\[ x = \sqrt{121} = 11 \quad \text{(since \( x \) must be positive)} \]

### Step 5: Use the identity for \( m - \frac{1}{m} \)
We know that:
\[ m^2 + \frac{1}{m^2} = \left(m - \frac{1}{m}\right)^2 + 2 \]
Let \( y = m - \frac{1}{m} \). Then we can express \( x \) as:
\[ x = y^2 + 2 \]

### Step 6: Substitute \( x \) back into the equation
Substituting \( x = 11 \):
\[ 11 = y^2 + 2 \]

### Step 7: Solve for \( y^2 \)
Now, we can solve for \( y^2 \):
\[ y^2 = 11 - 2 = 9 \]

### Step 8: Solve for \( y \)
Taking the square root of both sides, we find:
\[ y = \sqrt{9} = 3 \quad \text{(since \( y \) must be positive)} \]

### Final Answer
Thus, we have:
\[ m - \frac{1}{m} = 3 \]

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