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SOURCE:COMPETITION Number of Problems: 8. FOR PRINT ::: (Book)
What is the value of
Factoring out, we get:
Cancelling out the from the numerator and denominator, we see that it simplifies to .
Find the value of that satisfies the equation
Manipulate the powers of in order to get a clean expression.
If two numbers are equal, and their bases are equal, then their exponents are equal as well. Set the two exponents equal to each other.
Suppose that , , , and . What is ?
We can write as , as , as , and as . We know that can be rewritten as , so
What non-zero real value for satisfies ?
Taking the seventh root of both sides, we get .
Simplifying the LHS gives , which then simplifies to .
Thus, , and the answer is .
Real numbers and satisfy the equations and . What is ?
And
Substitution gives , and solving for yields . Thus .