Question

Using suitable identity, evaluate the following:

`((103)^3 - (3)^3)/((103)^2 + 103 xx 3 + (3)^2)`

Answer 1

Given expression: `((103)^3 - (3)^3)/((103)^2 + 103 xx 3 + (3)^2)`

Step-wise calculation using the identity for difference of cubes:

Recall the identity:

a³ – b³ = (a – b)(a² + ab + b²)

In the denominator, we have a² + ab + b², which corresponds to the term in the factorisation of the numerator.

Set:

a = 103, b = 3

Then:

a³ – b³ = (a – b)(a² + ab + b²)

Therefore:

`(a^3 - b^3)/(a^2 + ab + b^2) = ((a - b)(a^2 + ab + b^2))/(a^2 + ab + b^2)`

`(a^3 - b^3)/(a^2 + ab + b^2) = a - b`

Substitute back: 

103 – 3 = 100