Factorize :
`64a^(3)-1`

Answer

To factorize the expression \( 64a^3 - 1 \), we can use the difference of cubes formula, which states:

\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]

### Step 1: Identify \( a \) and \( b \)
In our case, we can rewrite \( 64a^3 \) and \( 1 \) as cubes:
- \( 64a^3 = (4a)^3 \)
- \( 1 = 1^3 \)

So, we have:
- \( a = 4a \)
- \( b = 1 \)

### Step 2: Apply the difference of cubes formula
Using the formula \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \), we substitute \( a \) and \( b \):

\[ 64a^3 - 1 = (4a)^3 - 1^3 = (4a - 1)((4a)^2 + (4a)(1) + (1)^2) \]

### Step 3: Simplify the expression
Now we simplify the second part of the expression:

1. Calculate \( (4a)^2 \):
\[ (4a)^2 = 16a^2 \]

2. Calculate \( (4a)(1) \):
\[ (4a)(1) = 4a \]

3. Calculate \( (1)^2 \):
\[ (1)^2 = 1 \]

Putting it all together, we have:
\[ (4a - 1)(16a^2 + 4a + 1) \]

### Final Answer
Thus, the factorization of \( 64a^3 - 1 \) is:
\[ (4a - 1)(16a^2 + 4a + 1) \]

---