Find the value of 27x³ + 8y³, if
(i) 3x + 2y = 14 and xy=8
(ii) 3x + 2y = 20 and xy=
Answers
Given : (i) 3x + 2y = 14 and xy = 8
(ii) 3x + 2y = 20 and xy = 14/9
To find : value of 27x³ + 8y³
Solution :
(i) 3x + 2y = 14 and xy = 8
By using the identity, (a + b)³ = a³ + b³ + 3ab(a + b)
On cubing 3x + 2y = 14 both sides,
(3x + 2y)³ = 14³
(3x)³ + (2y)³ + 3( 3x )(2y) (3x + 2y) = 2744
27x³ + 8y³ + 18xy(3x + 2y) = 2744
27x³ + 8y³ + 18 x 8 x 14 = 2744
[xy = 8]
27x³ + 8y³ + 2016 = 2744
27x³ + 8y³ = 2744 - 2016
27x³ + 8y³ = 728
Hence the value of 27x³ + 8y³ is 728.
(ii) 3x + 2y = 20 and xy = 14/9
By using the identity, (a + b)³ = a³ + b³ + 3ab(a + b)
On cubing 3x + 2y = 20 both sides,
(3x + 2y)³ = 20³
(3x)³ + (2y)³ + 3( 3x )(2y) (3x + 2y) = 8000
27x³ + 8y³ + 18xy(3x + 2y) = 8000
27x³ + 8y³ + 18 x 14/9 x 20 = 8000
[xy = 14/9]
27x³ + 8y³ + 560 = 8000
27x³ + 8y³ = 8000 - 560
27x³ + 8y³ = 7440
Hence the value of 27x³ + 8y³ is 7440.
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Answer:
Step-by-step explanation:
On cubing 3x + 2y = 14 both sides,
(3x + 2y)³ = 14³
(3x)³ + (2y)³ + 3( 3x )(2y) (3x + 2y) = 2744
27x³ + 8y³ + 18xy(3x + 2y) = 2744
27x³ + 8y³ + 18 x 8 x 14 = 2744
[xy = 8]
27x³ + 8y³ + 2016 = 2744
27x³ + 8y³ = 2744 - 2016
27x³ + 8y³ = 728