Factorise :
`a^(4) -1`

Answer

To factorise \( a^4 - 1 \), we can follow these steps:

### Step 1: Recognize the difference of squares
We can express \( a^4 - 1 \) as a difference of squares:
\[ a^4 - 1 = (a^2)^2 - (1)^2 \]

### Step 2: Apply the difference of squares formula
Using the formula \( x^2 - y^2 = (x - y)(x + y) \), we can factor \( a^4 - 1 \):
\[ a^4 - 1 = (a^2 - 1)(a^2 + 1) \]

### Step 3: Factor \( a^2 - 1 \) further
Notice that \( a^2 - 1 \) is also a difference of squares:
\[ a^2 - 1 = (a - 1)(a + 1) \]

### Step 4: Combine all factors
Now we can combine all the factors we have found:
\[ a^4 - 1 = (a^2 - 1)(a^2 + 1) = (a - 1)(a + 1)(a^2 + 1) \]

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

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