निम्नलिखित को सरल कीजिये :
(i) `(4x +2y)^(3) + ( 4x - 2y)^(3)`

Explanation

हमें ज्ञात है की : `(a+b)^(3) + (a-b)^(3)= {a^(3) +b^(3) + 3ab (a+b) } + { a^(3) -b^(3) - 3ab(a-b)}`
` rArr (a+b)^(3) + (a-b)^(3) = (a^(3) + b^(3) + 3a^(2) b+ 3ab^(2)) + (a^(3) b^(3) 3a^(2) b + 3ab^(2))`
` rArr (a+3)^(3) + (a-b)^(3) = 2(a^(3) + 3ab^(2))`
तथा
` (a+b)^(3) - (a-b)^(3) = { a^(3) +b^(3) +3ab (a+b)} - { a^(3) - b^(3) - 3ab (a-b)}`
` rArr (a+b)^(3) (a-b)^(3) = ( a^(3) +b^(3) +3a^(2)b+3ab^(2))- (a^(3) -b^(3) - 3a - 2b + 3ab^(2))`
` rArr (a+b)^(3) - (a-b)^(3) = a^(3) + b^(3) + 3a^(2) b + 3ab^(2) - a^(3) + b^(3) + 3a^(2)b - 3ab^(2)`
` rArr (a+b)^(3) - (a-b)^(3) = 2(b^(3) + 3a^(2)b)`
इन परिणामो का उपयोग करने पर
`(4x+2y)^(3) + (4x - 2y)^(3)`
` = 2 { (4x)^(3) +3 xx (4x) xx (2y)^(2)}" "` [ a = 4x तथा b=2y लेने पर ]
` = 2 (64x^(3) + 48xy^(2)) = 128x^(3) + 96xy^(2)`
वैकल्पिक विधि : `(4x+2y)^(3) + (4x-2y)^(3)`
` = { (4x)^(3) + (2y)^(3) + 3 xx 4x xx 2y (4x+2y)}`
` +{ (4x)^(3) +(-2y)^(3) + 3xx 4x xx 2y (4x + 2y)}`
` = { 64x^(3) + 8y^(3) + 24 xy (4x + 2y)} + { 64x^(3) - 8y^(3) - 24xy (4x-2y)}`
` = 64 x^(3) + 8y^(3) + 96x^(2)y + 48xy^(2) + 64 x^(3) - 8y^(3) - 96x^(2) y + 48xy^(2)`
` = 128x^(3) + 96 xy^(2)`