Question

¾X=27 what is the value of X
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Answer 1

Answer:

An obvious solution is x=3 . But are there others?

Note that, if x>y≥1 , then xx>yy . Thus, the function f(x)=xx is strictly increasing for x>1 , so xx=27 at no more than one point in this interval. And such a point indeed exists - namely, at x=3 .

If 0<x<1 , then xx<1x=1 , so xx≠27 everywhere in this interval.

If −1<x<0 , then (the principal branch of) xx is essentially taking the root of a negative number, which does not produce a real number.

If x≤−1 and xx is real, then |xx|=||x|x|=|1/|x||x|| . Since |x|>1 , this expression is less than 1, so this interval has no solutions.

Therefore, the only solution to xx=27 is x=3 .