Answer :
82) Answer:
[tex]\displaystyle \frac{1}{xy}[/tex]
Step-by-step explanation:
  We will simplify the given expressions by using exponent properties. The properties I have used are outlined below.
    [tex]\displaystyle \text{Exponent Product Property: } \boxed{a^m*a^n=a^{m+n}}[/tex]
    [tex]\displaystyle \text{Quotient Product Property: } \boxed{\frac{a^m}{a^n}=a^{m-n}}[/tex]
Given:
  [tex]\displaystyle (\frac{x^{\frac{3}{2}}}{x^2y^\frac{1}{2}})^2[/tex]
Rewrite squared into being multiplied by itself:
  [tex]\displaystyle (\frac{x^{\frac{3}{2}}}{x^2y^\frac{1}{2}})(\frac{x^{\frac{3}{2}}}{x^2y^\frac{1}{2}})[/tex]
Multiply fractions straight across:
➜ Numerator times numerator divided by denominator times denominator.
  [tex]\displaystyle (\frac{x^{\frac{3}{2}}*x^{\frac{3}{2}}}{x^2y^\frac{1}{2}*x^2y^\frac{1}{2}})[/tex]
Apply the exponent product property:
  [tex]\displaystyle (\frac{x^{\frac{3}{2}+\frac{3}{2}}}{x^{2+2}y^{\frac{1}{2}+\frac{1}{2}}})[/tex]
Addition:
  [tex]\displaystyle (\frac{x^{\frac{6}{2}}}{x^{4}y^{\frac{2}{2}}})[/tex]
Division:
  [tex]\displaystyle (\frac{x^{3}}{x^{4}y^{1}})[/tex]
Divide by applying the exponent quotient property:
➜ [tex]\frac{x^3}{x^4} =x^{3-4}=x^{-1}=\frac{1}{x}[/tex]
  [tex]\displaystyle \frac{1}{xy}[/tex]
83) Answer:
[tex]\displaystyle v^{\frac{9}{2}} u^{\frac{65}{24}[/tex]
Step-by-step explanation:
  We will simplify the given expressions by using exponent properties. The properties I have used are outlined below.
    [tex]\displaystyle \text{Exponent Product Property: } \boxed{a^m*a^n=a^{m+n}}[/tex]
    [tex]\displaystyle \text{Exponent Quotient Property: } \boxed{\frac{a^m}{a^n}=a^{m-n}}[/tex]
    [tex]\displaystyle \text{Exponent Power Property: } \boxed{(a^m)^n=a^{m*n}}[/tex]
Given:
[tex]\displaystyle \frac{u^{\frac{1}{3}}*(u^2v^3)^{\frac{7}{4}}}{(vu^{\frac{3}{2}})^{\frac{3}{4}}}[/tex]
Apply the exponent power property:
[tex]\displaystyle \frac{u^{\frac{1}{3}}*(u^{2*{\frac{7}{4}}}v^{3*{\frac{7}{4}}})}{(v^{1*{\frac{3}{4}}}u^{\frac{3}{2}*{\frac{3}{4}}})}[/tex]
Multiply fractions straight across:
➜ Numerator times numerator divided by denominator times denominator.
[tex]\displaystyle \frac{u^{\frac{1}{3}}*(u^{{\frac{7}{2}}}v^{{\frac{21}{4}}})}{(v^{{\frac{3}{4}}}u^{\frac{9}{8}})}[/tex]
Multiply the numerator by applying the exponent product property:
[tex]\displaystyle \frac{u^{{\frac{7}{2}+{\frac{1}{3}}}}v^{{\frac{21}{4}}}}{v^{{\frac{3}{4}}}u^{\frac{9}{8}}}[/tex]
Addition:
➜ [tex]\frac{7}{2} +\frac{1}{3} =\frac{21}{6} +\frac{2}{6} =\frac{23}{6}[/tex]
[tex]\displaystyle \frac{u^{{\frac{23}{6}}}v^{{\frac{21}{4}}}}{v^{{\frac{3}{4}}}u^{\frac{9}{8}}}[/tex]
Divide by applying the exponent quotient property:
[tex]\displaystyle v^{\frac{9}{2}} u^{\frac{65}{24}[/tex]
