If x = a^{m + n}, y = a^{n + l} and z = a^{l + m}, prove that x^myⁿz^l = xⁿy^lz^m.

Substituting values of x, y and z in L.H.S. of x^myⁿz^l = xⁿy^lz^m we get,

x^myⁿz^l = (a^{m + n})^m.(a^{n + l})ⁿ.(a^{l + m})^l

= a^{m2 + mn}.a^{n2 + ln}.a^{lm + l2}

= a^{m2 + n2 + l2 + mn + lm + ln} …….(i)

Substituting values of x, y and z in R.H.S. of x^myⁿz^l = xⁿy^lz^m we get,

xⁿy^lz^m = (a^{m + n})ⁿ.(a^{n + l})^l.(a^{l + m})^m

= a^{mn + n2}.a^{nl + l2}.a^{lm + m2}

= a^{m2 + n2 + l2 + mn + lm + ln} ………(ii)

Since (i) = (ii),

Hence, proved that x^myⁿz^l = xⁿy^lz^m.