how to factorise 81m⁴-(n-m)⁴ using the identities?
 

Dear Student,

Please find below the solution to the asked query :

$$\begin{gathered} 81{\mathrm{m}}^4 - {\left(\mathrm{n}-\mathrm{m}\right)}^4 \\ ={\left(9{\mathrm{m}}^2\right)}^2 - {\left\{{\left(\mathrm{n}-\mathrm{m}\right)}^2\right\}}^2 \\ =\left[\left(9{\mathrm{m}}^2\right)+{\left(\mathrm{n}-\mathrm{m}\right)}^2\right]\left[\left(9{\mathrm{m}}^2\right)-{\left(\mathrm{n}-\mathrm{m}\right)}^2\right] \\ =\left[\left(9{\mathrm{m}}^2\right)+{\left(\mathrm{n}-\mathrm{m}\right)}^2\right] \left[{\left(3\mathrm{m}\right)}^2 - {\left(\mathrm{n}-\mathrm{m}\right)}^2\right] \\ =\left(9{\mathrm{m}}^2+{\mathrm{n}}^2+{\mathrm{m}}^2-2\mathrm{mn}\right)\left[9{\mathrm{m}}^2-{\mathrm{n}}^2-{\mathrm{m}}^2+2\mathrm{mn}\right] \\ =\left(10{\mathrm{m}}^2+{\mathrm{n}}^2-2\mathrm{mn}\right)\left(8{\mathrm{m}}^2-{\mathrm{n}}^2+2\mathrm{mn}\right) \end{gathered}$$

Hope this would clear your doubt.

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