`27a^3+64b^3`

Answer

To factorize the expression \(27a^3 + 64b^3\), we can follow these steps:

### Step 1: Identify the cubes
First, we recognize that \(27a^3\) and \(64b^3\) can be expressed as cubes:
- \(27 = 3^3\), so \(27a^3 = (3a)^3\)
- \(64 = 4^3\), so \(64b^3 = (4b)^3\)

Thus, we can rewrite the expression:
\[ 27a^3 + 64b^3 = (3a)^3 + (4b)^3 \]

### Step 2: Apply the sum of cubes formula
We use the sum of cubes formula, which states:
\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \]
In our case, let \(x = 3a\) and \(y = 4b\). Therefore, we can apply the formula:
\[ (3a)^3 + (4b)^3 = (3a + 4b)((3a)^2 - (3a)(4b) + (4b)^2) \]

### Step 3: Calculate each part
Now we need to calculate each part of the formula:
1. \(x + y = 3a + 4b\)
2. \(x^2 = (3a)^2 = 9a^2\)
3. \(xy = (3a)(4b) = 12ab\)
4. \(y^2 = (4b)^2 = 16b^2\)

Putting these together, we have:
\[ (3a)^2 - (3a)(4b) + (4b)^2 = 9a^2 - 12ab + 16b^2 \]

### Step 4: Write the final factorized form
Now we can combine everything to write the final factorized expression:
\[ 27a^3 + 64b^3 = (3a + 4b)(9a^2 - 12ab + 16b^2) \]

### Final Answer
Thus, the factorization of \(27a^3 + 64b^3\) is:
\[ (3a + 4b)(9a^2 - 12ab + 16b^2) \]

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