Glossary Checklist FAQ

Use the exponent laws to complete the following exercises.

1. Simplify the following expressions.
   
   
   1. \(y^8  \div y^6\)
      
      
      Solution

      \[\begin{align}
       y^8  \div y^6 &= y^{8 - 6}  \\
        &= y^2  \\
       \end{align}\]

      
      
      
      
   2. \(\frac{{\left( {x^3 } \right)^2 }}{{x^6 }}\)
      
      
      Solution

      \[\begin{align}
       \frac{{\left( {x^3 } \right)^2 }}{{x^6 }} &= \frac{{x^{3\cdot 2} }}{{x^6 }} \\
        &= \frac{{x^6 }}{{x^6 }} \\
        &= x^{6 - 6}  \\
        &= x^0  \\
        &= 1 \\
       \end{align}
      \]

      
      
      
      
   3. \(\frac{{\left( {mn} \right)^2 }}{{m^3 n}}\)
      
      
   Solution
   \[\begin{align}
    \frac{{\left( {mn} \right)^2 }}{{m^3 n}} &= \frac{{m^2 n^2 }}{{m^3 n}} \\
     &= m^{2 - 3} n^{2 - 1}  \\
     &= m^{ - 1} n \\
     &= \frac{n}{m} \\
    \end{align}\]
   
   


2. Rewrite each radical as a power.
   
   
   1. \(\sqrt[3]{h^4}\)
      
      
      Solution

      \[\sqrt[3]{{h^4 }} = h^{\frac{4}{3}} \]

      
      
      
      
   2. \(\sqrt[6]{d^2}\)
      
      
      Solution

      \[\begin{align}
       \sqrt[6]{{d^2 }} &= d^{\frac{2}{6}}  \\
        &= d^{\frac{1}{3}}  \\
       \end{align}\]

      
      
      
      
3. Rewrite each power as a radical.
   
   
   1. \(t^{\frac{4}{5}} \)
      
      
      Solution

      \[t^{\frac{4}{5}}  = \sqrt[5]{{t^4 }}\]
      
      
      
      
   2. \(v^{\frac{1}{3}} \)
      
      
      Solution

      \[v^{\frac{1}{3}}  = \sqrt[3]{v}\]
      
      
      
4. Evaluate the following expressions without using a calculator.  Verify your answers using a calculator.
   
   
   1. \(\sqrt[3]{{64^2 }}\)
      
      
      Solution

      \[\begin{align}
       \sqrt[3]{{64^2 }} &= 64^{\frac{2}{3}}  \\
        &= \left( {4^3 } \right)^{\frac{2}{3}}  \\
        &= 4^{3\left( {\frac{2}{3}} \right)}  \\
        &= 4^2  \\
        &= 16 \\
       \end{align}\]
      

      
      
      
      
   2. \(\sqrt[5]{{32^4 }}\)
      
      
      Solution

      \[\begin{align}
       \sqrt[5]{{32^4 }} &= 32^{\frac{4}{5}}  \\
        &= \left( {2^5 } \right)^{\frac{4}{5}}  \\
        &= 2^{5\left( {\frac{4}{5}} \right)}  \\
        &= 2^4  \\
        &= 16 \\
       \end{align}\]