Question

Factorise the following:

`27x^3 + 1/(27x^3)`

Answer 1

Given the expression: `27x^3 + 1/(27x^3)`

Step-wise calculation:

1. Rewrite the terms as cubes:

27x³ = (3x)³ and `1/(27x^3) = (1/(3x))^3`

2. Recognize the expression as a sum of cubes:

`(3x)^3 + (1/(3x))^3`

3. Use the sum of cubes factorization formula:

a³ + b³ = (a + b)(a² – ab + b²) where a = 3x and `b = 1/(3x)`

4. Calculate each factor:

`a + b = 3x + 1/(3x)`

a² = (3x)²

a² = 9x²

`ab = (3x) xx 1/(3x)`

ab = 1

`b^2 = (1/(3x))^2`

`b^2 = 1/(9x^2)`

5. Substitute into the quadratic factor:

`a^2 - ab + b^2 = 9x^2 - 1 + 1/(9x^2)`

`27x^3 + 1/(27x^3) = (3x + 1/(3x))(9x^2 - 1 + 1/(9x^2))`