Concept
algebraic identities
(a+b)² = a²+b²+2ab
(a+b)³ = a³+b³+3ab(a+b)
Given
2 variable a and b such that a=5+2√6 and b=1/a
Find
we need to find the value of a^2+b^2 and a^3+b^3
Solution
We have
a = 5 +2√6
b = 1/(5+2√6)
Rationalizing b we get,
(5-2√6)/1 = 5-2√6
Thus,
a + b = 5 +2√6 + 5-2√6
= 10
and ab = (5 +2√6) (5-2√6)
= 25 - 24
= 1
Now,
(a+b)² = a²+b²+2ab
(10)² = a²+b²+ 2*1
100 = a²+b²+ 2
a²+b² = 98
Similarly
(a+b)³ = a³+b³+3ab(a+b)
(10)³ = a³+b³+3*1*10
1000 = a³+b³+30
a³+b³ = 970
Thus, a²+b² = 98 and a³+b³ = 970
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