If m and n are integers and x>0, what is value of x^m + x^n?

I concluded that statement 2 was sufficient.

This is what I did.

X^m + X^n = 1 + (X^n)/(X^m)

Statement 2: 1 + (X^n)*(X^-m) = 10 ---> 1 + (X^n)/(X^m) = 10

Hence: X^m + X^n = 1 + (X^n)/(X^m) = 10 ---> Sufficient.

Can anyone please explain me what am I doing wrong?

Thank you.

How did you get the highlighted part?

The question asks us for the value of: X^m + X^n

If we divide this expression by X^m, we get: 1 + (X^n)/(X^m)

Per statement 2: 1 + (X^n)/(X^m) = 10.

Hence, statement 2 allows us to know what is the value of the initial expression.

Where am I making the error?

I see what you are doing there but it's wrong. We need the value of x^m + x^n. When you divide this by x^m, you get the value of (x^m + x^n)/x^m, not x^m + x^n.

I choose C but I was wondering if my approach was correct?
Here is what I did :
1) insuff.
2) insuff.

C) Given 1+x^n−m=10 I rewrote as 1+ x^n/x^m = 10 then substitute x^m from 1)
I got 1+ x^n/81 = 10
81+x^n=810
x^n=729
So If I got x^n and x^m to answer the question.

But is it a correct way to arrive at solution?
thank you

This is how GMAT prep solved it and it makes sense:

Statement 1
X^m = 81 insufficient as there we need two equations for two variables (n & m), therefore insufficient

Statement 2
1 + x^n - m = 10
we can try to solve it and see what gives us
so... x^n-m = 10 -1
we get ...
x^n-m = 9
we can then separate both n-m by doing the reciprocals:
then we get x^n / x^m = 9. This is why statement 2 is not sufficient since we cannot find the value of "n". Therefore, not sufficient.

Now, we are left with statements 1 + 2 and see if the problem can be solved
Remember statement 1? x^m = 81

Remember your result from statement 2?

now you can replace x^m = 81 in your second statement where x^n / 81 = 9. Then we can take x^n = 9*81, which is 279.

Then we have as a result that x^n = 279 and x^m = 81. Now you only need to sum both.... 81 + 279 = 810. Therefore, the answer is "C" not because we got "810" but because the problem can be solved. This is what data sufficiency questions are asking you to determine...can the problem be solved? In this case only when both statements are combined the problem can be solved.

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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

(1) x can be 3 or 9:

\(3^4 = 81\)
\(9^2 = 81\)

We can't determine x^n, however. INSUFFICIENT.

(2) \(1 + \frac{x^n}{x^m} = 10\)
\(\frac{x^n}{x^m} = 9\)

We don't know the value of n or m; INSUFFICIENT.

(1&2) \(\frac{x^n}{81} = 9\)
\(x^n = 729\)

SUFFICIENT.

Answer is C.

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