c) If a -1/a =4, find the value i. a 2 + 1/a 2 ii. a 4 + 1/ a 4
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Step-by-step explanation:
Given :-
a - (1/a) = 4
To find :-
Find the following :
1) a²+(1/a²)
2) a⁴+(1/a⁴)
Solution :-
Given that : a-(1/a) = 4 ------------(1)
On squaring both sides then
=> [a-(1/a)]² = 4²
=> a²-2(a)(1/a)+(1/a)² = 16
Since (x-y)² = x²-2xy+y²
Where , x = a and y = 1/a
=> a²-2+(1/a²) = 16
=> a²+(1/a²) = 16-2
=> a²+(1/a²) = 14 ---------------(2)
On squaring both sides again ,then
=> [a²+(1/a²)]²= 14²
=> (a²)²+2(a²)(1/a²)+(1/a²)² = 196
Since (x+y)² = x²+2xy+y²
Where , x = a² and y = 1/a²
=> a⁴+2+(1/a⁴) = 196
=> a⁴+(1/a⁴) = 196-2
=> a⁴+(1/a⁴) = 194----------------(3)
Answer:-
The value of a²+(1/a²) is 14
The value of a⁴+(1/a⁴) is 194
Used formulae:-
- (x-y)² = x²-2xy+y²
- (x+y)² = x²+2xy+y²
- (a^m)^n = a^(mn)
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