If x = 4+√15 , find value of x^3 - 1/ x^3
Answers
Answer:
126√15
Step-by-step explanation:
ATQ , x = 4 + √15
1 / x = 1 / ( 4 + √15 )
Multiplying by ( 4 - √15 ) in numerator and denominator we get
= ( 4 - √15 ) / ( 4 + √ 15 ) ( 4 - √15 )
We have an identity as follows
a² - b² = ( a + b ) (a - b ) , applying it we get
= ( 4 - √15 ) / { ( 4 )² - ( √15 )² }
= ( 4 - √15 ) / ( 16 - 15 )
= ( 4 - √15 ) / 1
= ( 4 - √15 )
Now we find , value of
x - ( 1 / x ) = (4 + √15 ) - ( 4 - √15 )
= ( 4 + √15 - 4 + √15 )
= 2 √15
Now squaring both sides we get
( x - 1 / x )² = ( 2 √15 )²
We know that
( a - b )² = a² + b² - 2ab , applying this we get
=> (x)² + (1 / x)² - 2 ( x ) (1 / x ) = 4 × 15
=> x² + 1 / x² - 2 = 60
=> x² + 1 / x² = 60 +2
=> x² + 1 / x² = 62
Now returning to original problem
x³ - 1/x³ = ( x )³ - ( 1 / x )³
We have an identity
a³ - b³ = ( a - b ) (a² + b² + ab ) , using it we get
x³ - 1 / x³
= ( x - 1/x ) { x² + (1 / x²) + x (1 / x ) }
= ( 2 √15 ) { (x² + 1 / x² ) + 1 }
= ( 2√15 ) ( 62 + 1 )
= ( 2√15 ) ( 63 )
x³ - (1/x³ )= 126 √15
126√15
#answerwithquality #bal