Question

if x^2+1/x^2=7, find the value of x^3+1/x^3​

Answer 1

Answer:

(X+1/X)(X^2+1/X^2)

( X+1/X)=9

(X^2+1/X^2)=7

x^3+1/x^3 = 9x7= 56

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Answer 2

Answer:

( when x + 1/x = 3 )

• x³ + 1/x³ = 18

( when x + 1/x = -3 )

• x³ + 1/x³ = -18

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