Simplify the following expression: \[ \left(\frac{x^2}{x^m}\right)^{1+m} \left(\frac{x^m}{x^n}\right)^{m+n} \left(\frac{x^n}{x^l}\right)^{n+l} \]
1
Using the power rule \((x^a/x^b)^c = x^{(a-b)c}\), each factor simplifies to a single power of x.
The first factor becomes \(x^{(l-m)(l+m)} = x^{l^2-m^2}\), the second becomes \(x^{(m-n)(m+n)} = x^{m^2-n^2}\), and the third becomes \(x^{(n-l)(n+l)} = x^{n^2-l^2}\).
Multiplying these together adds the exponents: \((l^2-m^2)+(m^2-n^2)+(n^2-l^2) = 0\).
So the expression equals \(x^0 = 1\).
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