Salut,

if \(\displaystyle y = x^n\) and the curve passes between A(2,200) and B(2, 2000), first note that point B is higher than point A. So it means that the curve goes under B, and over A, at \(\displaystyle x = 2\). Therefore :

\(\displaystyle 200 < 2^n < 2000\)

You can then take the base-2 logarithm of each member of the inequality to extract \(\displaystyle n\) :

\(\displaystyle \log_2{(200)} < \log_2{(2^n)} < \log_2{(2000)}\)

\(\displaystyle 7.64 < n < 10.96\)

So :

\(\displaystyle 7 < n < 11\)

(by rounding, and noting that the question says that the curve passes between the points, not through the points).

Therefore the only options left are \(\displaystyle \boxed{n = 8}\), \(\displaystyle \boxed{n = 9}\) and \(\displaystyle \boxed{n = 10}\)

All those values of \(\displaystyle n\) satisfy your statement. So how do we know which one is the good one ?

**(!)** Look at the curve : the values are negative with \(\displaystyle x < 0\), and positive \(\displaystyle x > 0\). This is only possible if the exponent is odd (if it is even, the result is always positive). Therefore, the only solution is ... \(\displaystyle \boxed{n = 9}\) !

(Tongueout)