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If x^2+1/x^2=7, find the value of x^3+1/x^3 |
Answer» Answer:( when X + 1/x = 3 )
( when x + 1/x = -3 )
Step-by-step explanation:[Given that] → x² + 1/x² = 7 [ADDING 2 both sides to make LHS a perfect square] → x² + 1/x² + 2 = 7 + 2 [ ∵ (x) (1/x) = 1 ] → (x)² + (1/x)² + 2 (x) (1/x) = 9 [ using algebraic identity (a+b)² = a² + b² + 2 ab ] → ( x + 1/x )² = 9 → x + 1/x = ±3 ▶ Taking x + 1/x = 3 → x + 1/x = 3 [ cubing both sides ] → (x + 1/x)³ = 3³ [ using identity ( a + b )³ = a³ + b³ + 3 ab ( a + b ) ] → x³ + 1/x³ + 3 (x) (1/x) ( x +1/x) = 27 → x³ + 1/x³ + 3 ( 3 ) = 27 → x³ + 1/x³ = 27 - 9 → x³ + 1/x³ = 18 ▶ Taking x + 1/x = -3 → x + 1/x = -3 [ cubing both sides ] → ( x + 1/x )³ = (-3)³ → x³ + 1/x³ + 3 (x) (1/x) (x + 1/x) = -27 → x³ + 1/x³ + 3 ( - 3 ) = -27 → x³ + 1/x³ = -27 + 9 → x³ + 1/x³ = -18 |
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