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Question

if 3x+2y=12 and xy=6 then find the value of 27x3+8y3

Solution
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It is given that 3x+2y=12 and xy=6.

We find the value of 27x3+8y3 as shown below:

27x3+8y3=(3x)3+(2y)3

=(3x+2y)[(3x)2(3x×2y)+(2y)2](a³+b³=(a+b)(a²ab+b²))

=12[(3x)2(6×xy)+(2y)2](3x+2y=12,xy=6)

=12[(3x)2(6×6)+(2y)2]

=12[(3x)236+(2y)2]

=12[(3x)2+(2y)2+(2×3x×2y)(2×3x×2y)36]

=12[(3x+2y)212xy36]

=12[(12)2(12×6)36]

=12(1447236)

=12(144108)

=12×36

=432

Hence, 27x3+8y3=432.

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