(i) 27x3y3 – 8
Above terms can be written as,
(3xy)3 – 23
We know that, a3 – b3 = (a – b) (a2 + ab + b2)
Where, a = 3xy, b = 2
= (3xy – 2) ((3xy)2 + (3xy × 2) + 22)
= (3xy – 2) (9x2y2 + 6xy + 4)
(ii) 27(x + y)3 + 8(2x – y)3
Above terms can be written as,
33(x + y)3 + 23(2x – y)3
(3(x + y))3 + (2(x – y))3
We know that, a3 + b3 = (a + b) (a2 – ab + b2)
Where, a = 3(x + y), b = 2(x – y)
= [3(x + y) + 2(2x – y)] [(3(x + y))3 – (3(x + y) × 2(2x – y)) + (2(2x – y))2]
= [3x + 3y + 4x – 2y] [9(x + y)2 – 6(x + y)(2x – y) + 4(2x – y)2]
= (7x – y) [9(x2 + y2 + 2xy) – 6(2x2 – xy + 2xy – y2) + 4(4x2 + y2 – 4xy)]
= (7x – y) [9x2 + 9y2 + 18xy – 12x2 – 6xy – 6y2 + 16x2 + 4y2 – 16xy]
= (7x – y) [13x2 – 4xy + 19y2]
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