Factorising 24√3x^3-125y^3

To factorise the given expression, we can use the difference of cubes formula. The difference of cubes formula states that for any two numbers a and b, the expression a^3 - b^3 can be factorised as (a - b)(a^2 + ab + b^2).

In our case, we have 24√3x^3 - 125y^3, which can be rewritten as (2^3 * 3 * x^3) - (5^3 * y^3). Let's apply the difference of cubes formula to factorise it.

Step 1: Identify the cubes

The cubes in our expression are (2x)^3 and (5y)^3.

Step 2: Apply the difference of cubes formula

Using the difference of cubes formula, we can rewrite our expression as follows:

24√3x^3 - 125y^3 = (2x)^3 - (5y)^3

This can now be factorised as:

= [(2x) - (5y)][(2x)^2 + (2x)(5y) + (5y)^2]

Step 3: Simplify the factors

Now, let's simplify the factors:

= (2x - 5y)(4x^2 + 10xy + 25y^2)

Therefore, the factorised form of 24√3x^3 - 125y^3 is (2x - 5y)(4x^2 + 10xy + 25y^2).

Summary:
To factorise the expression 24√3x^3 - 125y^3, we used the difference of cubes formula. By identifying the cubes and applying the formula, we obtained the factorised form as (2x - 5y)(4x^2 + 10xy + 25y^2).