Part a)
Given the expression
[tex]54m^3n\:+\:81m^4n^2[/tex]
Apply exponent rule: [tex]a^{b+c}=a^ba^c[/tex]
[tex]54m^3n\:+\:81m^4n^2=54m^3n+81m^3mnn[/tex] ∵ [tex]m^4n^2=m^3mnn[/tex]
Rewrite 81 as 3 · 27
Rewrite 54 as 2 · 27
[tex]=2\cdot \:27m^3n+3\cdot \:27m^3mnn[/tex]
Factor out the common term: 27m³n
[tex]=27m^3n\left(2+3mn\right)[/tex]
Therefore,
[tex]54m^3n\:+\:81m^4n^2=54m^3n+81m^3mnn=27m^3n\left(2+3mn\right)[/tex]
Part B)
Given the expression
[tex]15x^2y^3z+25x^3y^2z+35x^2y^2z[/tex]
Apply exponent rule: [tex]a^{b+c}=a^ba^c[/tex]
[tex]15x^2y^3z+25x^3y^2z+35x^2y^2z=15x^2y^2yz+25x^2xy^2z+35x^2y^2z[/tex]
Rewrite as
[tex]=3\cdot \:5y^2x^2zy+5\cdot \:5y^2x^2zx+7\cdot \:5y^2x^2z[/tex]
Factor out common term 5y²x²z
[tex]=5y^2x^2z\left(3y+5x+7\right)[/tex]
Therefore,
[tex]15x^2y^3z+25x^3y^2z+35x^2y^2z=5y^2x^2z\left(3y+5x+7\right)[/tex]
