Question

x=a^m+n, y=a^n+l, z=a^l+m then show that, x^m×y^n×z^l=x^n×y^l×z^m​

Answer 1

Question

x=a^m+n, y=a^n+l, z=a^l+m then show that, x^m×y^n×z^l=x^n×y^l×z^m​

To Prove :

$x^m*y^n*z^l=x^n*y^l*z^m$ ​...(1)

Proof :

Given : $x=a^{m+n},y=a^{n+l},z=a^{l+m}$

Now sub. these values in (1) , we get ,

$\implies (a^{m+n})^m*a^{(n+l)}^n*(z^{l+m})^l=(a^{m+n})^n*(a^{n+l})^l*(a^{l+m})^m\\\\=>a^{m^2+mn}*a^{n^2+nl}*z^{l^2+lm}=a^{mn+m^2}*a^{nl+l^2}*a^{lm+m^2}\\\\=>a^{m^2+mn+n^2+nl+l^2+lm}=a^{m^2+mn+n^2+nl+l^2+lm}\\\\Since ,\:a^m.a^n=a^{m+n}\\\\=>\frac{a^{m^2+mn+n^2+nl+l^2+lm}}{a^{m^2+mn+n^2+nl+l^2+lm}}=1\\\\ =>1=1$

Since the values of both sides are same

So, $x^m*y^n*z^l=x^n*y^l*z^m$

Hence showed.