Question

if a= 2+√3 find the value of (a-1/a)² and (a+1/a)²​

Answer 1

Solution :

Given value of a :

a = 2 + √3.

Let us find the reciprocal of a or 1/a.

1/a

=> 1/( 2 + √3 )

=> ( 2-√3)/( 2+√3)(2-√3)

=> 2-√3

Now ,

a + 1/a

=> 2 + √3 + 2 - √3

=> 4

a - 1/a

=> 2 + √3 - (2-√3)

=> 2 + √3 - 2 + √3

=> 2√3

(a + 1/a)² = ( 4)² = 16

( a - 1/a)² = ( 2√3) = 12 .

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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