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| A Value Problem (Posted on 2007-01-14) |
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Given that, H3- H -1 = 0.
Determine analytically the exact value of:
(3H2 - 4H)1/3 + H(2H2+3H+2)1/4
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Submitted by K Sengupta
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Rating: 4.0000 (1 votes)
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Solution:
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(Hide)
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By the problem;
H^3 - H - 1 = 0
Or, H = H^3 - 1
Or, H+1 = H^3
So, 3*H^2 - 4H
= 3*H^2 - 3H - (H^3 -1)
= 1 - 3H + 3*H^2 - H^3
= (1-H)^3
Now, 2*H^2 +3H +2
= H^2 + (H^2 + H) + 2(H+1)
= H^2 + H(H^3) + 2*H^3
= H^4 + 2*H^3 + H^2
= (H^2)[(H+1)^2]
= H^2*H^6
= H^8
Therefore, the given expression is
= 1 - H + H^3
= 1 - H + 1 + H
= 2
Consequently, the required value of the expression is 2.
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