Question

If `x = a^(m + n), y = a^(n + l)` and `z = a^(l + m),` prove that `x^my^nz^l = x^ny^lz^m`

Answer 1

Given `x = a^(m + n), y = a^(n + l)` and `z = a^(l + m)`

Putting the values of x, y and z in `x^my^nz^l,` we get

`x^my^nz^l`

= `(a^(m + n))^m(a^(n + l))^n(a^(l + m))^l`

= `(a^(m^2 + nm))(a^(n^2 + ln))(a^(l^2 + lm))`

= `a^(m^2 + n^2 + l^2 + nm + ln + lm)`

Putting the values of x, y and z in `x^ny^lz^m,` we get

`x^ny^lz^m`

= `(a^(m + n))^n(a^(n + l))^l(a^(l + m))^m`

= `(a^(mn + n^2))(a^(nl + l^2))(a^(lm + m^2))`

= `a^(mn+n^2 + nl + l^2 + lm + m^2)`

So, `x^my^nz^l = x^ny^lz^m`.