`216x^3+(1)/(125)`

Answer

To factorize the expression \( 216x^3 + \frac{1}{125} \), we can follow these steps:

### Step 1: Rewrite the expression in terms of cubes
We start by recognizing that \( 216x^3 \) and \( \frac{1}{125} \) can be expressed as cubes:
\[ 216x^3 = (6x)^3 \quad \text{and} \quad \frac{1}{125} = \left(\frac{1}{5}\right)^3 \]
Thus, we can rewrite the expression as:
\[ (6x)^3 + \left(\frac{1}{5}\right)^3 \]

### Step 2: Apply the sum of cubes formula
The sum of cubes can be factored using the formula:
\[ A^3 + B^3 = (A + B)(A^2 - AB + B^2) \]
Here, let \( A = 6x \) and \( B = \frac{1}{5} \). We can now apply the formula:
\[ (6x)^3 + \left(\frac{1}{5}\right)^3 = (6x + \frac{1}{5})\left((6x)^2 - (6x)(\frac{1}{5}) + \left(\frac{1}{5}\right)^2\right) \]

### Step 3: Calculate \( A^2 \), \( AB \), and \( B^2 \)
Now we need to calculate each component:
- \( A^2 = (6x)^2 = 36x^2 \)
- \( AB = (6x)(\frac{1}{5}) = \frac{6x}{5} \)
- \( B^2 = \left(\frac{1}{5}\right)^2 = \frac{1}{25} \)

### Step 4: Substitute back into the formula
Substituting these values back into the expression gives us:
\[ (6x + \frac{1}{5})\left(36x^2 - \frac{6x}{5} + \frac{1}{25}\right) \]

### Step 5: Simplify the second bracket
To simplify the second bracket, we can find a common denominator (which is 25):
\[ 36x^2 = \frac{900x^2}{25}, \quad -\frac{6x}{5} = -\frac{30x}{25}, \quad \text{and} \quad \frac{1}{25} = \frac{1}{25} \]
Thus, we have:
\[ 36x^2 - \frac{6x}{5} + \frac{1}{25} = \frac{900x^2 - 30x + 1}{25} \]

### Step 6: Final expression
Putting it all together, the factorized form of the original expression is:
\[ (6x + \frac{1}{5})\left(\frac{900x^2 - 30x + 1}{25}\right) \]
We can also express this as:
\[ \frac{1}{25}(6x + \frac{1}{5})(900x^2 - 30x + 1) \]

### Final Answer
The complete factorization is:
\[ \frac{1}{25}(6x + \frac{1}{5})(900x^2 - 30x + 1) \]