Question

`8x^3 - 27/y^3` is equal to ______.

Options

• `(2x - 3/y)(4x^2 + 6xy + 9/y^2)`

• `(2x + 3/y)(4x^2 - (6x)/y + 9/y^2)`

• `(2x - 3/y)(4x^2 + (6x)/y - 9/y^2)`

• `(2x - 3/y)(4x^2 + (6x)/y - 9/y^2)`

Answer 1

`8x^3 - 27/y^3` is equal to `underlinebb((2x - 3/y)(4x^2 + (6x)/y - 9/y^2))`.

Explanation:

We are given the expression:

`8x^3 - 27/y^3`

Step 1: Recognise it as a difference of cubes.

This is a difference of cubes since:

`8x^3 = (2x)^3` and `27/y^3 = (3/y)^3`

Thus, we can rewrite the expression as:

`(2x)^3 - (3/y)^3`

Step 2: Apply the difference of cubes formula.

The difference of cubes formula is:

a³ – b³ = (a – b)(a² + ab + b²)

Here, `a = 2x` and `b = 3/y`.

Using the formula:

`(2x)^3 - (3/y)^3 = (2x - 3/y)((2x)^2 + (2x) * 3/y + (3/y)^2)`

Step 3: Simplify the terms inside the second bracket.

• `(2x)^2 = 4x^2`
• `(2x) * 3/y = (6x)/y`
• `(3/y)^2 = 9/y^2`

Thus, the factorised expression is`(2x - 3/y)(4x^2 + (6x)/y + 9/y^2)`