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Question: For ![](https://cdn.pureessence.tech/canvas_359.png?top_left_x=353&top_left_y=1026&width=300&height=...

For (xmxn)(m2+nm+n2)(xnxl)(n2+nl+l2)\left( \frac{x^{m}}{x^{n}} \right)^{(m^{2} + nm + n^{2})}\left( \frac{x^{n}}{x^{l}} \right)^{(n^{2} + nl + l^{2})}

A

1

B

x

C

Does not exist

D

None of these

Answer

1

Explanation

Solution

(xlxm)l2+lm+m2(xmxn)m2+nm+n2(xnxl)n2+nl+l2\left( \frac{x^{l}}{x^{m}} \right)^{l^{2} + lm + m^{2}}\left( \frac{x^{m}}{x^{n}} \right)^{m^{2} + nm + n^{2}}\left( \frac{x^{n}}{x^{l}} \right)^{n^{2} + nl + l^{2}}

=(xl−m)(l2+lm+m2)(xm−n)m2+nm+n2(xn−l)n2+nl+l2(x^{l - m})^{(l^{2} + lm + m^{2})}(x^{m - n})^{m^{2} + nm + n^{2}}(x^{n - l})^{n^{2} + nl + l^{2}}

=xl3−m3.xm3−n3.xn3−l3x^{l^{3} - m^{3}}.x^{m^{3} - n^{3}}.x^{n^{3} - l^{3}}=xl3−m3+m3−n3+n3−l3=x0x^{l^{3} - m^{3} + m^{3} - n^{3} + n^{3} - l^{3}} = x^{0}=1