Factorise :
`a^(4) - 343 a`

Answer

To factorise the expression \( a^4 - 343a \), we can follow these steps:

### Step 1: Factor out the common term
The first step is to identify and factor out the common term from the expression. In this case, both terms in the expression \( a^4 \) and \( -343a \) have a common factor of \( a \).

\[ a^4 - 343a = a(a^3 - 343) \]

### Step 2: Recognize the difference of cubes
Next, we notice that \( 343 \) can be expressed as \( 7^3 \). Therefore, we can rewrite the expression inside the parentheses as a difference of cubes:

\[ a^3 - 7^3 \]

### Step 3: Apply the difference of cubes formula
The difference of cubes can be factored using the formula:

\[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) \]

In our case, \( x = a \) and \( y = 7 \). So we apply the formula:

\[ a^3 - 7^3 = (a - 7)(a^2 + 7a + 49) \]

### Step 4: Combine the factors
Now we can combine all the factors we have found:

\[ a(a^3 - 7^3) = a(a - 7)(a^2 + 7a + 49) \]

### Final Answer
Thus, the completely factored form of the expression \( a^4 - 343a \) is:

\[ \boxed{a(a - 7)(a^2 + 7a + 49)} \]

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