Question

if x^10+1/x^10 =3 than x^50+1/x^50=?​

Answer 1

Solution :

Given that ;

x¹⁰ + 1/x¹⁰ = 3

Squaring this :

( x¹⁰ + 1/x¹⁰)² = 9

=> x²⁰ + 1/x²⁰ + 2 = 9

=> x²⁰ + 1/x²⁰ = 7 ....... (1)

x¹⁰ + 1/x¹⁰ = 3

Cubing this ;

( x¹⁰ + 1/x¹⁰ )³ = 3³

=> x³⁰ + 1/³⁰ + 2 = 27

=> x³⁰ + 1/x³⁰ = 25. ....... (2)

(1) × (2)

=> ( x²⁰ + 1/x¹⁰)( x³⁰ + 1/x³⁰) = 25 × 7 = 175

=> x⁵⁰ + 1/x⁵⁰ + 3 = 175

=> x⁵⁰ + 1/x⁵⁰ = 172 .

This is the required answer .

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Additional Information :

(a + b)² = a² + 2ab + b²

(a + b)² = (a - b)² + 4ab

(a - b)² = a² - 2ab + b²

(a - b)² = (a + b)² - 4ab

a² + b² = (a + b)² - 2ab

a² + b² = (a - b)² + 2ab

2 (a² + b²) = (a + b)² + (a - b)²

4ab = (a + b)² - (a - b)²

ab = {(a + b)/2}² - {(a-b)/2}²

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

(a + b)³ = a³ + 3a²b + 3ab² b³

(a + b)³ = a³ + b³ + 3ab(a + b)

(a - b)³ = a³ - 3a²b + 3ab² - b³

a³ + b³ = (a + b)( a² - ab + b² )

a³ + b³ = (a + b)³ - 3ab( a + b)

a³ - b³ = (a - b)( a² + ab + b²)

a³ - b³ = (a - b)³ + 3ab ( a - b )

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