(1). if x= ( 1 +√2) , find the value of (x - 1/x)³.
(2). If (3+√7)/(3-√7) = a+b√7 , find the value of a and b.
Answers
(x - 1/x)³ = x³ - 1/x³ - 3x + 3/x
= (1+√2)³ - 1/(1+√2)³ - 3(1 + √2) + 3/(1+√2)
(1 + √2)³ = 1 + 2√2 + 3√2 + 6 = 7 + 5√2 ..... (1)
1/(1+√2)³ = 1 = 7 - 5√2 = 7 - 5√2 = - 7 + 5√2 .....(2)
7 + 5√2 7² - (5√2)² 49 - 50
3(1 + √2) = 3+ 3√2 ..............(3)
3/(1+ √2) = 3 = 3 - 3√2 = - 3 + 3√2 ........(4)
1 + √2 1 - 2
(1+√2)³ - 1/(1+√2)³ - 3(1 + √2) + 3/(1+√2)
= (7 + 5√2) - ( - 7 + 5√2) - ( 3+ 3√2) + (- 3 + 3√2)
= 7 + 5√2 + 7 - 5√2 - 3 - 3√2 - 3 + 3√2
= 14 - 6
= 8
2.
(3 + √7 ) = (3 + √7)² = 9 + 7 + 6√7 = 16+ 6√7 = 8 + 3√7
(3- √7) 9 - 7 2 2
a = 8
b = 3√7
At last finished