Evaluate this expression, using pen and paper only:
3√(1 - 18*(3)
1/3 + 12* 3
2/3)
Let t = 3^(1/3) so I don't have to keep typing 3^(1/3).
I'm using keyboard and Notepad instead of pen and paper.
We have (12t^2 - 18t + 1)^(1/3)
Let 12t^2 - 18t + 1 = (a^(1/3) + b^(1/3))^3
12t^2 - 18t + 1 = a + b + 3a^(2/3)b^(1/3) + 3a^(1/3)b^(2/3)
Setting these to be equivalent gives a system of equations
a + b = 1
3a^(2/3)b^(1/3) = -18t
3a^(1/3)b^(2/3) = 12t^2
Divide the second by the third
(a/b)^(1/3) = - 3/(2t)
a/b = - 27/(8t^3) but t = 3^(1/3)
a/b = -27/24 = -9/8
8a = -9b and a+b = 1
8(1-b) = -9b
b = -8 and a = 9
a^(1/3) + b^(1/3) = 3^(2/3) - 2 (the solution)
Cube it to check
(3^(2/3) - 2)^3 = 9 - 6*3^(4/3) + 12*3^(2/3) - 8
= 1 - 18*3^(1/3) + 12*3^(2/3)
The solution is 3^(2/3) - 2
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Posted by Larry
on 2023-10-20 08:58:59 |
Solution
