Given that, H³- H -1 = 0.

Determine analytically the exact value of:

(3H² - 4H)^1/3 + H(2H²+3H+2)^1/4

I really enjoyed solving this, so thank you. The answer is 2.

From the given equation, I take two separate identities that I use a lot later: H = H³ - 1 and 1 = H³ - H

First let's look at just 3H² - 4H.
3H² - 4H = 3H² - 3H - H = 3H² - 3H - H + H³ - H³ = (H³ - H) - 3H + 3H² - H³ = 1 - 3H + 3H² - H³ = (1 - H)³
Therefore (3H² - 4H)^1/3 = ((1 - H)³)^1/3 = 1 - H

And now we'll solve the second bit:
2H² + 3H + 2 = 2H² + 3H + 2(H³ - H) = 2H³ + 2H² + H = H(2H² + 2H + 1) = H(2H² + 2H + H³ - H) = H²(H² + 2H + 1) = H²(H + 1)²
Therefore, H(2H² + 3H + 2)^1/4 = H(H²(H + 1)²)^1/4 = H * H^1/2 * (H + 1)^1/2 = H^3/2 * (H + 1)^1/2 = (H³)^1/2 * (H + 1)^1/2 = (H + 1)^1/2 * (H + 1)^1/2 = H + 1

And finally, (3H² - 4H)^1/3 + H(2H² + 3H + 2)^1/4 = 1 - H + 1 + H = 2

I know, unnecessarily long and complicated. But it was so much fun to play with the algebra. ^_^

Posted by TamTam on 2007-01-16 10:11:55