1970 AHSME Problems/Problem 13
Problem
Given the binary operation defined by for all positive numbers and . Then for all positive , we have
Solution
Let . If all of them are false, the answer must be . If one does not fail, we will try to prove it.
For option , we have , which is clearly false.
For option , we have , which is false.
For option , we have , which is false.
For option , we have , which is true.
The LHS is . By the elementary definition of exponentiation, this is multiplied by itself times. Since each is actually multiplied times, the expression is multiplied by itself times.
The RHS is . This is multiplied by itself times.
Thus, the LHS is always equal to the RHS, so is the only correct statement.
See also
1970 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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