1990 AIME Problems/Problem 2
Find the value of .
Suppose that is in the form of . FOILing yields that . This implies that and equal one of . The possible sets are and ; the latter can be discarded since the square root must be positive. This means that . Repeating this for , the only feasible possibility is .
The power is quite irritating to work with so we look for a way to eliminate that. Notice that squaring the expression will accomplish that. Let be the sum of the given expression. After doing the arithmetic (note that the first two terms will have some cancellation and that the last term will simplify quickly using difference of squares), we arrive at which gives .
Factor as a difference of cubes. We can simplify the left factor as follows. Since , we know that , so our final answer is .
Let , . Similarly to solution 2, we let The expression can be simplified as follow Thus .
(Similar to Solution 3, but with substitution)
Let and We want to find
We have Then,
Our answer is
(Similar to Solution 1, but expanding the cubes instead)
Like in Solution 1, we have and
Therefore we have that
From here, we use the formula and . Applying them to our problem we get that We see that all the terms with square roots cancel, leaving us with
Note: We have that because we need the square root to be positive and since is obviously greater than So we have
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