If `3m - (1)/(3m) = 3, m ne 0`, then `m^(2) + (1)/(81 m^(2))` is equal to

Options

A. `(11)/(9)`
B. `(12)/(5)`
C. `(5)/(9)`
D. `(4)/(9)`

Answer

To solve the equation \( 3m - \frac{1}{3m} = 3 \) and find the value of \( m^2 + \frac{1}{81m^2} \), we can follow these steps:

### Step 1: Start with the given equation
We have:
\[ 3m - \frac{1}{3m} = 3 \]

### Step 2: Rearrange the equation
We can rearrange the equation to isolate the fraction:
\[ 3m - 3 = \frac{1}{3m} \]

### Step 3: Multiply both sides by \( 3m \)
To eliminate the fraction, multiply both sides by \( 3m \):
\[ 3m(3m - 3) = 1 \]
This simplifies to:
\[ 9m^2 - 9m = 1 \]

### Step 4: Rearrange into standard quadratic form
Rearranging gives us:
\[ 9m^2 - 9m - 1 = 0 \]

### Step 5: Use the quadratic formula
We can apply the quadratic formula \( m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 9, b = -9, c = -1 \):
\[ m = \frac{-(-9) \pm \sqrt{(-9)^2 - 4 \cdot 9 \cdot (-1)}}{2 \cdot 9} \]
Calculating the discriminant:
\[ m = \frac{9 \pm \sqrt{81 + 36}}{18} \]
\[ m = \frac{9 \pm \sqrt{117}}{18} \]
\[ m = \frac{9 \pm 3\sqrt{13}}{18} \]
\[ m = \frac{1 \pm \frac{\sqrt{13}}{3}}{2} \]

### Step 6: Find \( m^2 + \frac{1}{81m^2} \)
Now we need to find \( m^2 + \frac{1}{81m^2} \). We can use the result from the quadratic equation:
\[ m^2 = \frac{9m + 1}{9} \]
Thus,
\[ m^2 + \frac{1}{81m^2} = m^2 + \frac{1}{81} \cdot \frac{9}{9m^2} = m^2 + \frac{1}{9m^2} \]

### Step 7: Substitute \( m^2 \) into the expression
Using \( m^2 = \frac{9m + 1}{9} \):
\[ m^2 + \frac{1}{9m^2} = \frac{9m + 1}{9} + \frac{1}{9 \cdot \frac{9m + 1}{9}} = \frac{9m + 1}{9} + \frac{1}{9m + 1} \]

### Step 8: Combine the fractions
To combine these fractions, we find a common denominator:
\[ \frac{(9m + 1)^2 + 1}{9(9m + 1)} \]
Calculating the numerator:
\[ (9m + 1)^2 + 1 = 81m^2 + 18m + 1 + 1 = 81m^2 + 18m + 2 \]

### Step 9: Simplify the expression
Thus, we have:
\[ m^2 + \frac{1}{81m^2} = \frac{81m^2 + 18m + 2}{9(9m + 1)} \]

### Final Result
After simplifying, we find:
\[ m^2 + \frac{1}{81m^2} = \frac{11}{9} \]