Factorize: `3sqrt(3)a^3-b^3-5sqrt(5)c^3-3sqrt(15)a b c`

Answer

To factorize the expression \(3\sqrt{3}a^3 - b^3 - 5\sqrt{5}c^3 - 3\sqrt{15}abc\), we can follow these steps:

### Step 1: Identify the structure
We can recognize that the expression resembles the form of \(x^3 + y^3 + z^3 - 3xyz\). We will rewrite our expression in this form.

### Step 2: Rewrite the expression
We can rewrite the expression as:
\[ 3\sqrt{3}a^3 + (-b^3) + (-5\sqrt{5}c^3) - 3\sqrt{15}abc \]
Here, we can identify:
- \(x = \sqrt{3}a\)
- \(y = -b\)
- \(z = -\sqrt{5}c\)

### Step 3: Apply the factorization formula
Using the formula \(x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)\), we can substitute \(x\), \(y\), and \(z\) into the formula.

### Step 4: Calculate \(x + y + z\)
\[ x + y + z = \sqrt{3}a - b - \sqrt{5}c \]

### Step 5: Calculate \(x^2 + y^2 + z^2\)
\[ x^2 = (\sqrt{3}a)^2 = 3a^2 \]
\[ y^2 = (-b)^2 = b^2 \]
\[ z^2 = (-\sqrt{5}c)^2 = 5c^2 \]
Thus,
\[ x^2 + y^2 + z^2 = 3a^2 + b^2 + 5c^2 \]

### Step 6: Calculate \(xy + yz + zx\)
\[ xy = (\sqrt{3}a)(-b) = -\sqrt{3}ab \]
\[ yz = (-b)(-\sqrt{5}c) = \sqrt{5}bc \]
\[ zx = (-\sqrt{5}c)(\sqrt{3}a) = -\sqrt{15}ac \]
Thus,
\[ xy + yz + zx = -\sqrt{3}ab + \sqrt{5}bc - \sqrt{15}ac \]

### Step 7: Combine the results
Now we can combine everything:
\[ x^2 + y^2 + z^2 - xy - yz - zx = (3a^2 + b^2 + 5c^2) - (-\sqrt{3}ab + \sqrt{5}bc - \sqrt{15}ac) \]
This simplifies to:
\[ 3a^2 + b^2 + 5c^2 + \sqrt{3}ab - \sqrt{5}bc + \sqrt{15}ac \]

### Final Factorization
Thus, the factorization of the original expression is:
\[ ( \sqrt{3}a - b - \sqrt{5}c )( 3a^2 + b^2 + 5c^2 + \sqrt{3}ab - \sqrt{5}bc + \sqrt{15}ac ) \]