without actually calculating the cubes, find the value of 48^3 - 30^3 - 18^3.?

48³ - 30³ - 18³
= 48³ - 18³ - 30³
= (48³ - 18³) - 30³
= (48 - 18)(48² + 48×18 + 18²) - 30³ …… As a³ - b³ = (a - b)(a² + ab + b²)
= 30 × (48² + 48×18 + 18²) - 30³
= 30 × [(48² + 48×18 + 18²) - 30²]
= 30 × [(48² - 30²) + 48×18 + 18²]
= 30 × [(48 + 30)(48 - 30) + 48×18 + 18²] …… As a² - b² = (a + b)(a - b)
= 30 × [78 × 18 + 48×18 + 18²]
= 30 × 18 × 144
= 77760

= 48³ - 30³ - 18³

= (6 * 8)³ - (6 * 5)³ - (6 * 3)³

= (6³ * 8³) - (6³ * 5³) - (6³ * 3³)

= 6³ * [8³ - 5³ - 3³]

= 6³ * [512 - 125 - 27]

= 6³ * [(500 + 12) - (100 + 25) - (30 - 3)]

= 6³ * [500 + 12 - 100 - 25 - 30 + 3]

= 6³ * [500 - 100 - 25 - 30 + 12 + 3]

= 6³ * [400 - 55 + 15]

= 6³ * [400 - 40]

= 6³ * 360

= 6³ * (36 * 10)

= 6 * 6² * (36 * 10)


= 6 * 36 * 36 * 10

= 36² * 6 * 10

= [30 + 6]² * 6 * 10

= [30² + (2 * 30 * 6) + 6²] * 6 * 10

= [900 + (2 * 3 * 10 * 6) + 36] * 6 * 10

= [900 + (6 * 60) + 36] * 6 * 10

= [900 + 360 + 36] * 6 * 10

= (900 + 300 + 60 + 30 + 6) * 6 * 10

= (5400 + 1800 + 360 + 180 + 36) * 10

= (7200 + 360 + 180 + 36) * 10

= (7200 + 300 + 60 + 100 + 80 + 36) * 10

= (7200 + 400 + 60 + 80 + 36) * 10

= (7200 + 400 + 140 + 36) * 10

= (7200 + 400 + 100 + 40 + 36) * 10

= (7700 + 40 + 36) * 10

= (7700 + 76) * 10

= 7776 * 10

= 77760

I would look at it as
6^3 (8^3 - 5^3 - 3^3)
= 6^3 (512 - 125 - 27)
= 6^3 (512 - 152)
= 6^3 (360) = 216*360 = 77760

77760

48^3 - 30^3 - 18^3
= 83592 - 5832
= 77,760