Factorise :
`a^(3) + 0.064`

Answer

To factorise the expression \( a^3 + 0.064 \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ a^3 + 0.064 \]
We can express \( 0.064 \) as a fraction:
\[ 0.064 = \frac{64}{1000} \]

### Step 2: Recognize the cubes
Next, we can identify \( 64 \) and \( 1000 \) as perfect cubes:
\[ 64 = 4^3 \quad \text{and} \quad 1000 = 10^3 \]
Thus, we can rewrite \( 0.064 \) as:
\[ 0.064 = \left(\frac{4}{10}\right)^3 \]

### Step 3: Apply the sum of cubes formula
Now we can rewrite the expression as:
\[ a^3 + \left(\frac{4}{10}\right)^3 \]
This fits the sum of cubes formula, which states:
\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \]
Here, let \( x = a \) and \( y = \frac{4}{10} \).

### Step 4: Factor using the formula
Applying the formula:
\[ a^3 + \left(\frac{4}{10}\right)^3 = \left(a + \frac{4}{10}\right)\left(a^2 - a \cdot \frac{4}{10} + \left(\frac{4}{10}\right)^2\right) \]

### Step 5: Simplify the expression
Now we simplify the second factor:
1. Calculate \( \left(\frac{4}{10}\right)^2 = \frac{16}{100} = 0.16 \)
2. The second factor becomes:
\[ a^2 - a \cdot \frac{4}{10} + 0.16 = a^2 - 0.4a + 0.16 \]

### Final Result
Thus, the factorization of \( a^3 + 0.064 \) is:
\[ \left(a + 0.4\right)\left(a^2 - 0.4a + 0.16\right) \]

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