If `3x-(4)/(x)=4`, find `27x^(3)-(64)/(x^(3))`

Answer

To solve the equation \( 3x - \frac{4}{x} = 4 \) and find the value of \( 27x^3 - \frac{64}{x^3} \), we can follow these steps:

### Step 1: Rearranging the original equation
We start with the equation:
\[ 3x - \frac{4}{x} = 4 \]
We can rearrange this to isolate \( 3x \):
\[ 3x = 4 + \frac{4}{x} \]

### Step 2: Cubing both sides
Next, we will cube both sides of the equation to use the formula \( (a - b)^3 = a^3 - b^3 - 3ab(a - b) \). Here, let \( a = 3x \) and \( b = \frac{4}{x} \):
\[ (3x - \frac{4}{x})^3 = 4^3 \]
Thus,
\[ (3x - \frac{4}{x})^3 = 64 \]

### Step 3: Applying the formula
Using the formula:
\[ (3x)^3 - \left(\frac{4}{x}\right)^3 - 3(3x)\left(\frac{4}{x}\right)(3x - \frac{4}{x}) = 64 \]
Calculating \( (3x)^3 \) and \( \left(\frac{4}{x}\right)^3 \):
\[ 27x^3 - \frac{64}{x^3} - 3(3x)\left(\frac{4}{x}\right)(3x - \frac{4}{x}) = 64 \]

### Step 4: Simplifying the equation
Now, we need to simplify \( 3(3x)\left(\frac{4}{x}\right)(3x - \frac{4}{x}) \):
\[ 3(3x)\left(\frac{4}{x}\right)(3x - \frac{4}{x}) = 36(3x - \frac{4}{x}) \]
Substituting \( 3x - \frac{4}{x} = 4 \):
\[ 36(4) = 144 \]

### Step 5: Substituting back into the equation
Now we can substitute back into the equation:
\[ 27x^3 - \frac{64}{x^3} - 144 = 64 \]

### Step 6: Isolating \( 27x^3 - \frac{64}{x^3} \)
Rearranging gives:
\[ 27x^3 - \frac{64}{x^3} = 64 + 144 \]
\[ 27x^3 - \frac{64}{x^3} = 208 \]

### Final Answer
Thus, the value of \( 27x^3 - \frac{64}{x^3} \) is:
\[ \boxed{208} \]