Question

# Add the following: ${\mathrm{l}}^{2}+{\mathrm{m}}^{2},{\mathrm{m}}^{2}+{\mathrm{n}}^{2},{\mathrm{n}}^{2}+{\mathrm{l}}^{2},2\mathrm{lm}+2\mathrm{mn}+2\mathrm{nl}$

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Solution

## We have, ${\mathrm{l}}^{2}+{\mathrm{m}}^{2},{\mathrm{m}}^{2}+{\mathrm{n}}^{2},{\mathrm{n}}^{2}+{\mathrm{l}}^{2},2\mathrm{lm}+2\mathrm{mn}+2\mathrm{nl}$$\left({\mathrm{l}}^{2}+{\mathrm{m}}^{2}\right)+\left({\mathrm{m}}^{2}+{\mathrm{n}}^{2}\right)+\left({\mathrm{n}}^{2}+{\mathrm{l}}^{2}\right)+\left(2\mathrm{lm}+2\mathrm{mn}+2\mathrm{nl}\right)\phantom{\rule{0ex}{0ex}}={\mathrm{l}}^{2}+{\mathrm{m}}^{2}+{\mathrm{m}}^{2}+{\mathrm{n}}^{2}+{\mathrm{n}}^{2}+{\mathrm{l}}^{2}+2\mathrm{lm}+2\mathrm{mn}+2\mathrm{nl}\phantom{\rule{0ex}{0ex}}={\mathrm{l}}^{2}+{\mathrm{l}}^{2}+{\mathrm{m}}^{2}+{\mathrm{m}}^{2}+{\mathrm{n}}^{2}+{\mathrm{n}}^{2}+2\mathrm{lm}+2\mathrm{mn}+2\mathrm{nl}\left[\mathrm{Combining}\mathrm{like}\mathrm{terms}\mathrm{and}\mathrm{simplifying}\right]\phantom{\rule{0ex}{0ex}}=2{\mathrm{l}}^{2}+2{\mathrm{m}}^{2}+2{\mathrm{n}}^{2}+2\mathrm{lm}+2\mathrm{mn}+2\mathrm{nl}$Hence, when we add ${\mathrm{l}}^{2}+{\mathrm{m}}^{2},{\mathrm{m}}^{2}+{\mathrm{n}}^{2},{\mathrm{n}}^{2}+{\mathrm{l}}^{2},2\mathrm{lm}+2\mathrm{mn}+2\mathrm{nl}$ we get $2{\mathrm{l}}^{2}+2{\mathrm{m}}^{2}+2{\mathrm{n}}^{2}+2\mathrm{lm}+2\mathrm{mn}+2\mathrm{nl}$.

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