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If x is not equal to zero, and x+1/x = 3

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If x is not equal to zero, and x+1/x = 3  [#permalink]

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New post 18 Mar 2018, 12:13
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A
B
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Difficulty:

  75% (hard)

Question Stats:

59% (02:08) correct 41% (03:01) wrong based on 63 sessions

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If x is not equal to zero, and x+1/x = 3, then what is the value of x^6 + (1/x)^6?

A. 320
B. 322
C. 324
D. 326
E. 328
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If x is not equal to zero, and x+1/x = 3  [#permalink]

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New post 18 Mar 2018, 12:52
2
prady2231 wrote:
If x is not equal to zero, and x+1/x = 3, then what is the value of x^6 + (1/x)^6?

A. 320
B. 322
C. 324
D. 326
E. 328


Given: (\(x + \frac{1}{x}\)) = 3

Squaring on both sides, we get \((x + \frac{1}{x})^2 = 3^2\) -> \(x^2 +(\frac{1}{x^2}) + 2 = 9\) -> \(x^2 +(\frac{1}{x^2}) = 7\)

Applying cube on both sides, we get \((x^2 +(\frac{1}{x^2}))^3 = 7^3\)

Formula used: \((a + b)^3 = a^3 + b^3 + 3ab(a + b)\)

The equation simplifies to \((x^2)^3 + (\frac{1}{x^2})^3 + 3*x^2*\frac{1}{x^2}(x^2 +(\frac{1}{x^2})) = 343\)

Therefore, the value of \(x^6 + (\frac{1}{x^6}) + 3*7 = 343\) -> \(x^6 + (\frac{1}{x^6}) = 343 - 21 = 322\)(Option B)
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Re: If x is not equal to zero, and x+1/x = 3  [#permalink]

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New post 19 Mar 2018, 10:28
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Re: If x is not equal to zero, and x+1/x = 3  [#permalink]

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New post 03 Apr 2018, 03:00
My approach(I found it easy with squaring and proceeding):-
Given: x+1/x = 3
=> x^2 + (1/x)^2 + 2*x*(1/x) = 9
=> x^2 + (1/x)^2 = 7......(1)

Now, square eqn 1:-
x^4 + (1/x)^4 = 47.....(2)

Now multiply eqn 1 & 2:-
=> x^6 + (1/x)^2 + x^2 + (1/x)^6 = 329;
=> x^6 + 7 + (1/x)^6 = 329.....got "7" using eqn 1
=> x^6 + (1/x)^6 = 322

Hence, B is our answer.
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Re: If x is not equal to zero, and x+1/x = 3  [#permalink]

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New post 03 Apr 2018, 09:01
chandra004 wrote:
My approach(I found it easy with squaring and proceeding):-
Given: x+1/x = 3
=> x^2 + (1/x)^2 + 2*x*(1/x) = 9
=> x^2 + (1/x)^2 = 7......(1)

Now, square eqn 1:-
x^4 + (1/x)^4 = 47.....(2)

Now multiply eqn 1 & 2:-
=> x^6 + (1/x)^2 + x^2 + (1/x)^6 = 329;
=> x^6 + 7 + (1/x)^6 = 329.....got "7" using eqn 1
=> x^6 + (1/x)^6 = 322

Hence, B is our answer.

Possible, but pushpitkc has shown the fastest possible approach for exam hall situation, it is best to adopt that method.
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Re: If x is not equal to zero, and x+1/x = 3 &nbs [#permalink] 03 Apr 2018, 09:01
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