Practice Solutions

Simplify. Write the answer in radical form.

1. $\sqrt[]{8a^4{b}^9}$
   
   $$\begin{gathered} \sqrt{2^3 a^4 b^9} \\ {2}^{\frac{3}{2} }{a}^{\frac{4}{2} }{b}^{\frac{9}{2}}=2^{1\frac{1}{2}} a^{2 }{b}^{4\frac{1}{2}} \\ 2a^{2 }{b}^4 \sqrt[]{2b} \end{gathered}$$

2. $\sqrt[3]{8a^4{b}^9}$
   $$\begin{gathered} \sqrt[3]{2^3 a^4 b^9} \\ {2}^{\frac{3}{3} }{a}^{\frac{4}{3} }{b}^{\frac{9}{3}}=2^1 {a^{1\frac{1}{3}}}^{ }{b}^3 \\ 2a{b^3}^{ } \sqrt[3]{a} \end{gathered}$$

3. $\sqrt[]{25x^8{y}^{12}}$
   
   $$\begin{gathered} \sqrt{5^2 x^8 y^{12}} \\ {5}^{\frac{2}{2}} x^{\frac{8}{2}} y^{\frac{12}{2}}=5^1 x^4 y^6 \\ 5x^4{y}^6 \end{gathered}$$
   
   

4. $\sqrt[3]{25x^8{y}^{12}}$
   
   $$\begin{gathered} \sqrt[3]{5^2 x^8 y^{12}} \\ {5}^{\frac{2}{3}} x^{\frac{8}{3}} y^{\frac{12}{3}}=5^{\frac{2}{3}} x^{2\frac{2}{3}} y^4 \\ {x}^2{y}^4 \sqrt[3]{5^2{x}^2} \\ {x}^2{y}^4 \sqrt[3]{25x^2} \end{gathered}$$
   
   

5. $\sqrt[]{x^{22} y^{35} z^{40}}$
   $$\begin{gathered} x^{\frac{22}{2}} y^{\frac{35}{2}} z^{\frac{40}{2}}=x^{11} y^{17\frac{1}{2} }{z}^{20} \\ {x}^{11}{y}^{17}{z}^{20} \sqrt[]{y} \end{gathered}$$
   
   

6. $\sqrt[3]{x^{22} y^{35} z^{40}}$
   $$\begin{gathered} x^{\frac{22}{3}} y^{\frac{35}{3}} z^{\frac{40}{3}}=x^{7\frac{1}{3}} y^{11\frac{2}{3} }{z}^{13\frac{1}{3}} \\ {x}^7{y}^{11}{z}^{13} \sqrt[3]{x{y}^{2}z} \end{gathered}$$

 

Simplify. Write the answer in radical form.

7. $\sqrt[]{27 x^7 y^5 z^6}$
   
   $$\begin{gathered} \sqrt{3^3 x^7 y^5 z^6} \\ {3}^{\frac{3}{2}} x^{\frac{7}{2}} y^{\frac{5}{2}} z^{\frac{6}{2}}=3^{1\frac{1}{2}} x^{3\frac{1}{2} }{y}^{2\frac{1}{2} }{z}^3 \\ 3x^3{y}^2{z}^3 \sqrt[]{3xy} \end{gathered}$$
   
   

8. $\sqrt[3]{27 x^7 y^5 z^6}$
   
   $$\begin{gathered} \sqrt[3]{3^3 x^7 y^5 z^6} \\ {3}^{\frac{3}{3}} x^{\frac{7}{3}} y^{\frac{5}{3}} z^{\frac{6}{3}}=3^1 x^{2\frac{1}{3} }{y}^{1\frac{2}{3} }{z}^2 \\ 3x^{2}y{z}^2 \sqrt[3]{x{y}^2} \end{gathered}$$
   
   

9. $\sqrt[]{a^{18} b^{25} c^{15}}$
   $$\begin{gathered} a^{\frac{18}{2}} b^{\frac{25}{2}} c^{\frac{15}{2}}=a^9 b^{12\frac{1}{2}} c^{7\frac{1}{2}} \\ {a}^9 b^{12} c^7 \sqrt[]{bc} \end{gathered}$$
   
   

10. $\sqrt[3]{a^{18} b^{25} c^{15}}$
    $$\begin{gathered} a^{\frac{18}{3}} b^{\frac{25}{3}} c^{\frac{15}{3}}=a^6 b^{8\frac{1}{3}} c^5 \\ {a}^6 b^8 c^5 \sqrt[3]{b} \end{gathered}$$
    
    

11. $\sqrt[]{48x^5}$
    $$\begin{gathered} \sqrt[]{2^4 ·3^1 x^5} \\ {2}^{\frac{4}{2}} 3^{\frac{1}{2}} x^{\frac{5}{2}}=2^2 3^{\frac{1}{2}} x^{2\frac{1}{2}} \\ {2}^2 x^2 \sqrt[]{3x} \\ 4x^2\sqrt{3x} \end{gathered}$$
    
    

12. $\sqrt[3]{48x^5}$

    $$\begin{gathered} \sqrt[3]{2^4 ·3^1 x^5} \\ {2}^{\frac{4}{3}} 3^{\frac{1}{3}} x^{\frac{5}{3}}=2^{1\frac{1}{3}} 3^{\frac{1}{3}} x^{1\frac{2}{3}} \\ 2 x \sqrt[3]{2·3x^2} \\ 2 x \sqrt[3]{6x^2} \end{gathered}$$