(a+1/a)^2=3 find a^3+1/a^3
with method
Answers
Answer:
0
Step-by-step explanation:
Given--->
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(a+1/a)²=3
To find---> a³ + (1/a³)
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Solution---> We know an identity
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(a + b)² =a² + b² +2 a b
Applying this identity here
(a + 1/a )²= a² + 1/a² +2 a (1/a)
3 = a² + 1/ a² + 2
(a² + 1/a²) = 3 -2
(a² + 1/a²) = 1
Now
(a +1/a)² = 3
Taking square root of both sides
a + 1/a =√3
Now we have to find
a³ + 1/a³
We have one more identity
x³+y³=(x+y) (x²+y²-xy)
Applying this identity
a³ + 1/a³ =(a+1/a) {a² + 1/a² -a ×(1/a)}
a is cancel out from numerator & denominator
= (√3) {(a² + 1/a²) - 1 }
= (√3) ( 1 - 1)
= √3 (0)
= 0
Addional information--->
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1) (a - b)² =a² + b² -2ab
2)a³- b³ =(a - b ) (a² + b² +ab)
3)a² - b² = (a + b) (a - b)
4)(a+b+c)²= a²+ b² + c² + 2ab + 2bc + 2ca
5)(a + b)³ = a³ + b³ +3a b (a + b)
6)(a - b)³ =a³ - b³ - 3a b (a - b)
0
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