hii
if x = 2 + √3
find x² + 1/x² and x³ + 1/x³
plzz, it is branliest question...
Answers
Answered by
63
Hi there !
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Given :
x = 2 + √3
To Find :
(i) x² + 1 / x²
(ii) x³ + 1 / x³
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x = 2 + √3
1 / x = 1 / 2 + √3
1 / x = 1 / 2 + √3 × 2 - √3 / 2 - √3
1 / x = 2 - √3 / (2)² - (√3)²
1 / x = 2 - √3 / 4 - 3
1 / x = 2 - √3
Now,
x + 1 / x = 2 + √3 + 2 - √3
x + 1 / x = 2 + 2
x + 1 / x = 4
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Question I :
x + 1/x = 4
On squaring both sides ..
( x + 1/x )² = (4)²
x² + 1 / x² + 2 = 16
x² + 1 / x² = 16 - 2
x² + 1 / x² = 14 [Ans]
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Question II :
x + 1 / x = 4
On cubing both sides ..
( x + 1/x )³ = (4)³
x³ + 1/x³ + 3 ( x + 1/x) = 64
x³ + 1/x³ + 3 (4) = 64
x³ + 1/x³ + 12 = 64
x³ + 1/x³ = 64 - 12
x³ + 1/x³ = 52 [Ans]
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Thanks for the question !
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janu4878:
Sorry
Answered by
47
Heya!!
If x = 2 + √3
=) 1/x = 1/(2+√3) * (2-√3)/2-√3
= 2-√3 / (2^2 - √3^2)
= 2-√3
Since x^2 + 1/x^2
= (x + 1/x)^2 - 2*x*1/x
= (x +1/x)^2 - 2
= (2 + √3 + 2 - √3) ^2 - 2
= 4^2 - 2
= 16-2
= 14
2) x^3 + 1/x^3
= (2 + √3) ^3 + (2-√3)^2
= 8 + 3√3 + 12√3 + 18 + 8 - 3√3 - 12√3 + 18
= 52
Hope it helps uh!!
If x = 2 + √3
=) 1/x = 1/(2+√3) * (2-√3)/2-√3
= 2-√3 / (2^2 - √3^2)
= 2-√3
Since x^2 + 1/x^2
= (x + 1/x)^2 - 2*x*1/x
= (x +1/x)^2 - 2
= (2 + √3 + 2 - √3) ^2 - 2
= 4^2 - 2
= 16-2
= 14
2) x^3 + 1/x^3
= (2 + √3) ^3 + (2-√3)^2
= 8 + 3√3 + 12√3 + 18 + 8 - 3√3 - 12√3 + 18
= 52
Hope it helps uh!!
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