Question

If (a^2+1/a^2)= 27, find the value of: (I) (a^3+1/a^3)​

Answer 1

Answer:

Step-by-step explanation:

Step-by-step explanation:

Given---> a² + 1 / a² = 27

To find ---> value of ( a³ - 1 / a³ ).

Solution----> We know that,

( x - y )² = x² + y² - 2xy

Applying it for x = a and y = 1 / a , we get,

=> ( a - 1 / a )² = a² + 1 / a² - 2 × a × 1 / a

=> ( a - 1 / a )² = a² + 1 / a² - 2

=> ( a - 1 / a )² = ( a² + 1 / a² ) - 2

=> ( a - 1 / a )² = 27 - 2

=> ( a - 1 / a )² = 25

=> ( a - 1 / a ) = 5

We have an identity as follows,

( a³ - b³ ) = ( a - b ) ( a² + b² + ab )

Now ,

( a³ - 1 / a³ ) = ( a )³ - ( 1 / a )³

= ( a - 1 / a ) ( a² + 1 / a² + a × 1 / a )

= ( 5 ) { ( a² + 1 / a² ) + 1 }

= ( 5 ) { ( 27 ) + 1 }

= ( 5 ) ( 28 )

= 140