(a + b)3 = a3 + 3a2b + 3ab2 + b3
(a - b)3 = a3 - 3a2b + 3ab2 - b3
(a3 + b3) = (a + b)3 - 3ab(a + b)
(a3 - b3) = (a - b)3 - 3ab(a - b)
Example 1 :
Find x3 - y3, if x - y = 5 and xy = 14
Solution :
x3 - y3, if x - y = 5 and xy = 14
(a3 - b3) = (a - b)3 + 3ab(a - b)
(x3 - y3) = (x - y)3 + 3xy(x - y)
By using the given values, we get
(x3 - y3) = 53 + 3(14)(5)
= 125 + 210
= 335
Example 2 :
If a + (1/a) = 6, then find the value of a3 + 1/a3
Solution :
(a3 + b3) = (a + b)3 - 3ab(a + b)
a3 + (1/a)3 = (a + (1/a))3 - 3a(1/a)(a + (1/a))
= 63 - 3(6)
= 216 - 18
= 198
Example 3 :
If x2 + 1/x2 = 23, then find the value of x + (1/x) and x3 + (1/x3)
Solution :
a2 + b2 = (a + b)2 - 2ab
x2 + (1/x)2 = (x + (1/x))2 - 2x(1/x)
23 = (x + (1/x))2 - 2
23 + 2 = (x + (1/x))2
(x + (1/x))2 = 25
x + (1/x) = 5
x3 + (1/x)3 = (x + (1/x))3 - 3x(1/x)(x + (1/x))
= 53 - 3(5)
= 125 - 15
= 105
Hence the values of x + (1/x) and x3 + (1/x)3 are 5 and 105 respectively.
Example 4 :
If (y - (1/y))3 = 27, then find the value of y3 - (1/y)3
Solution :
(a3 - b3) = (a - b)3 - 3ab(a - b)
Given that :
(y - (1/y))3 = 27
(y - (1/y))3 = 33
(y - (1/y)) = 3
(y3 - (1/y)3) = (y - (1/y))3 - 3y(1/y)(y - (1/y))
= 27 - 3(3)
= 27 - 9
= 18
Hence the value of (y3 - (1/y)3) is 18.
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