In our first equation, we can notice:
[tex] (x^m*x^2)=x^{2+m} [/tex]
and
[tex] (x^{m+2})^3*k^{15}=x^{21} *k^{15} \implies \\
x^{m+2}=x^{21} [/tex]
So, using the fact that [tex] x^{y}=x^z \implies y=z [/tex], we have:
[tex] m+2=21 \implies\\
m=19 [/tex]
In our second equation, we have:
[tex] (x^3*y^2)(\frac{x^2y^3*z^m}{z^{-5}})=x^5y^5z [/tex]
So, using the exponent rules, we get:
[tex] (x^3*y^2)(\frac{x^2y^3*z^m}{z^{-5}})=\\ x^{2+3}*y^{3+2}*z^{m-(-5)}=\\x^5*y^5*z^{m+5} [/tex]
So, we can cross out the common factors in each to give us:
[tex] z^{m+5}=z \implies\\
z^{m+5}=z^1 \implies \\
m+5=1 \implies \\
m=-4 [/tex]
So, using the exponent rules, we found the m values in both equations.
