Given:
[tex]$x^{2} y^{8} \cdot \frac{x^{3} y^{?}}{x y^{3}}=x^{4} y^{2}[/tex]
To find:
The missing value.
Solution:
Let the missing value be m.
[tex]$x^{2} y^{8} \cdot \frac{x^{3} y^{m}}{x y^{3}}=x^{4} y^{2}[/tex]
Using exponent rule: [tex]a^m \cdot a^n=a^{m+n}[/tex]
[tex]$ \frac{x^{2+3} y^{8+m}}{x y^{3}}=x^{4} y^{2}[/tex]
[tex]$ \frac{x^{5} y^{8+m}}{x y^{3}}=x^{4} y^{2}[/tex]
Using exponent rule: [tex]\frac{a^m}{a^n} =a^{m-n}[/tex]
[tex]$ x^{5-1} y^{8+m-3}=x^{4} y^{2}[/tex]
[tex]$ x^{4} y^{5+m}=x^{4} y^{2}[/tex]
Equate x terms and y terms.
[tex]x^4=x^4[/tex] and [tex]y^{5+m}=y^2[/tex]
If the bases are same, then the powers are equal.
[tex]5+m=2[/tex]
Subtract 5 from both sides.
[tex]5+m-5=2-5[/tex]
[tex]m=-3[/tex]
The missing value is -3.
