Question

if 5x - 4y =7 and xy = 8, find : 125xcube - 64ycube

Answer 1

Solution:

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Given:


=> 5x - 4y = 7,


=> xy = 8,

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To find :


=> 125x³ - 64y³ =?

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if 5x - 4y =7,.


=> (5x - 4y)² = (7)²                       squaring both the sides


=> (5x)² - 2(5x)(4y) + (4y)² = (7)²


=> 25x² - 40xy + 16y² = 49


=> 25x² -40(8) +16y²= 49


=> 25x²- 320 + 16y² = 49


=> 25x² + 16y² =49 + 320


=> 25x² + 16y² =369,...(i)

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We can change the equation like this too,


=> 125x³ - 64y³


=> (5x)³ - (4y)³


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As, we know the identity that,.


=> a³ - b³ = (a - b)(a² + ab + b²)

 
let,


=> a = 5x ,


and


=> b = 4y,.


then, use the identity,.


=> (5x)³ - (4y)³ = (5x - 4y) ((5x)² + (5x) (4y) + (4y)²)


=> 125x³ - 64y³ = (7) (25x² +20xy + 16y²)


=> 125x³ - 64y³ = 7 (25x² + 16y² + 20(8))


=> 125x³ - 64y³ =  7 (369 + 160)


=> 125x³ - 64y³ = 7 (529)


=> 125x³ - 64y³ = 3703,..

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                                                 Hope it Helps !!