A2853. Find the product of the real solutions to (x

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A2853. Find the product of the real solutions to (x² −7x+ 11)^(x²^−13x+42) = 1.

Solution by the Missouri State University Problem Solving Group Department of Mathematics

Missouri State University 901 S. National Ave.

Springfield, MO 65897

Letf(x) =x²−7x+ 11 and g(x) =x²−13x+ 42.

We have f(x) = 1 if and only if (x−2)(x−5) = 0, and f(2)^g(2) = 1²⁰ = 1, f(5)^g(5) = 1² = 1.

So we have two solutions x= 2 andx= 5.

Now note that f(x) = −1 if and only if (x−3)(x−4) = 0, and f(3)^g(3) = (−1)¹² = 1, f(4)^g(4) = (−1)⁶ = 1, so we have two more solutions, x= 3 andx= 4.

Finally g(x) = 0 if and only if (x−6)(x−7) = 0 and f(6)^g(6) = 5⁰ = 1, f(7)^g(7) = 11⁰ = 1, so we have solutions x= 6 andx= 7.

The product of the real solutions is then 2·3·4·5·6·7 = 5040.

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