Question

Find the value of 64 x cube minus 125 y cube if 4 x minus 5 Y is equal to 16 and xy is equal to 12

Answer 1

$\underline{ \underline{ \bold{ \red {\: \: SOLUTION \: \: }}}}$➡

$\bold{\underline{Given \: :} \: \: \: 4x - 5y = 16} \\ \\ \bold{ \: \: \:\: \: \: \: \: \: \: \: \: \: \: \: \: \:xy = 12} \\ \\ \\ \bold{ \underline{To \: \: find \: : }\: \: 64x {}^{3} - 125y {}^{3} =? }$

$\bold{We \: \: know \: \: that} \\ \\ \boxed{\bold{a {}^{3} - b {}^{3} = (a - b)(a {}^{2} + ab + b {}^{2})} } \\ \\ \bold{ Again,} \\ \\ \bold{(a - b) {}^{2} + 3ab = a {}^{2} - 2ab + b {}^{2} + 3ab} \\ \\ \bold{ = >\underline{( a - b) {}^{2} + 3ab = a {}^{2} + ab + b {}^{2}} } \\ \\ \bold{So,} \\ \\ \boxed{\bold{a {}^{3} - b {}^{3} = (a - b)[(a - b) {}^{2} + 3ab]} }$

$\bold{Here,} \\ \\ \underline{\bold{We \: \: use \: \: the \: \: above \: \: formula \: \: for}} \\ \underline {\bold{solve \: \: the \: \: question \: \: }}$

$\bold{Now,} \\ \\ \bold{(4x - 5y) {}^{2} = (16) {}^{2} =256 } \\ \\ \bold{3 \timse 4x \times 5y =60xy= 60 \times 12 =720} \\ \\ \bold{Hence,} \\ \\ \bold{64x {}^{3} - 125y {}^{3} = (4x) {}^{3} - (5y) {}^{3} } \\ \\ \bold{ = (4x - 5y)[(4x - 5y) {}^{2} + 60xy]} \\ \\ \bold{ = 16 \times (256 + 720)} \\ \\ \bold{ = 16 \times 976} \\ \\ \bold{ = 15616}$


$\underline{ \underline{ \bold{ \red{ \: \: ANSWER \: \: }}}}$➡

$\boxed{ \boxed{ \bold{ \purple{ \: \: 64x {}^{3} - 125y {}^{3} = 4672 \: \: }}}}$

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✝$\mathfrak{ \red{ \: \: THANKS... \: \: }}$✝

Answer 2

SOLUTION:-

here,

4x-5y =16 and xy=12

we know that

a3-b3 = (a-b)(a2+ab+b2)

again,

(a-b)2 +3ab=a2-2ab+b2+3ab

(a-b)2+3ab=a2+ab+b2

so,

a3-b3=(a-b)[(a-b)2+3ab]

here,

we use the above formula for solving the equation,

now,

(4x-5y)2=(16)2=256

34x×5y=60xy=60×12=720

hence,

64x3-125y3=(4x)3-(5y)3

=(4x-5y)[(4x-5y)2+60xy]

=16×(256+720)

=16×976

=15616

64x3-125y3=4672