newbornmuse
If m and n are integers and x > 0, what is the value of \(x^m + x^n\)?
(1) \(x^m = 81\)
(2) \(1 + x^{n-m} = 10\)
Alternative approcach: \(NUMBER PICKING\)
given m and n are integers, and x is positive. we need to find the value of \(x^m + x^n\)?
the goal of number picking approach is try to prove insufficiency.
the statement will be insufficient if there are different values of the prompt question. if we are ending up with only one value (consistently) satisfying the statement then that will prove sufficiency.
statement 1: given \(x^m = 81\),
since m and n are integer and x can not be negative then there are only two possibilities exist
either x=9 and y=2
or x=3 and y=4
clearly, we can see that there will be different values of the prompt question \(x^m + x^n\)? as there are different values for the x and y.
thus INSUFFICIENT.
statement 2: gives \(1 + x^{n-m} = 10\)
\(1 + x^{n-m} = 10\)
= \(x^{n-m} = 9\)
= \(x^{n-m} = 3^{2} or (-3)^{2}\)
since x can not be negative then there is only one possibility
that is x=3 and (n-m)=2
but here we can have different values of n and m individually. for instance, n=6 and m=4 or n=8 and m=6 etc.
clearly, we can see that there will be different values of n and m and thus different values of the prompt question \(x^m + x^n\)?
so, INSUFFICIENT
now together, we need to satisfy the both statements together.
here the only common case between both statements is x=3. so if x=3 then from statement (1), we know m has to be 4. and if m has to be 4, from the statement (2) we know n must be 6, because (n-m)=2
so there is only one case and thus there should be only one values of the prompt question.
NOW, we do not need to find the value. all we have to know is whether there are multiple value exist or one value exist for the prompt question.
thus together we can answer the question that there is only one value fro the prompt question.
Hope this will be helpful