1.4.3 Exercises

Calculate the following powers:

1. ${\displaystyle {(-\frac{3}{5})}^4}$$\hspace{0.5em}=\hspace{0.5em}$ .
   Solution
   
   
   ${\displaystyle {(-\frac{3}{5})}^4=\frac{3^4}{5^4}=\frac{81}{625}\hspace{0.5em}.}$
   
   
   
   
2. ${\displaystyle {\left(2^{-2}\right)}^{-3}}$$\hspace{0.5em}=\hspace{0.5em}$ .
   Solution
   
   
   ${\displaystyle {\left(2^{-2}\right)}^{-3}=2^{(-2)·(-3)}=2^6=64\hspace{0.5em}.}$
   
   
   
   
3. ${\displaystyle {(-\frac{1}{2})}^0}$$\hspace{0.5em}=\hspace{0.5em}$ .
   Solution
   
   
   ${\displaystyle {(-\frac{1}{2})}^0=1\hspace{0.5em}.}$
   
   
   
   

Simplify the following expressions using the exponent rules and by reducing the results. You don't need to evaluate the powers:

1. ${\displaystyle \frac{(-2{)}^7}{(-2{)}^5}}$$\hspace{0.5em}=\hspace{0.5em}$ .
   Solution
   
   
   ${\displaystyle \frac{(-2{)}^7}{(-2{)}^5}=(-2{)}^{7-5}=(-2{)}^2\hspace{0.5em}.}$
   
   
   
   
2. ${\displaystyle 6^2·3^{-2}}$$\hspace{0.5em}=\hspace{0.5em}$ .
   Solution
   
   
   ${\displaystyle 6^2·3^{-2}=6^2·{\left(\frac{1}{3}\right)}^2={(6·\frac{1}{3})}^2=2^2\hspace{0.5em}.}$
   
   
   
   
3. ${\displaystyle \frac{{64}^3}{8^3}}$$\hspace{0.5em}=\hspace{0.5em}$ .
   Solution
   
   
   ${\displaystyle \frac{{64}^3}{8^3}={\left(\frac{64}{8}\right)}^3=8^3\hspace{0.5em}.}$
   
   
   
   
4. ${\displaystyle {\left(\frac{3}{4}\right)}^{\frac{1}{3}}·{\left(\frac{3}{4}\right)}^{\frac{2}{3}}}$$\hspace{0.5em}=\hspace{0.5em}$ .
   Solution
   
   
   ${\displaystyle {\left(\frac{3}{4}\right)}^{\frac{1}{3}}·{\left(\frac{3}{4}\right)}^{\frac{2}{3}}={\left(\frac{3}{4}\right)}^{\frac{1}{3}+\frac{2}{3}}\mathrm{\hspace{0.5em}\hspace{0.5em}}=\mathrm{\hspace{0.5em}\hspace{0.5em}}\frac{3}{4}\hspace{0.5em}.}$
   
   
   
   

Calculate the following roots (here, the result is always an integer):

1. ${\displaystyle \sqrt[3]{3}·\sqrt[3]{9}}$$\hspace{0.5em}=\hspace{0.5em}$ .
   Solution
   
   
   ${\displaystyle \sqrt[3]{3}·\sqrt[3]{9}=\sqrt[3]{3·3·3}=3\hspace{0.5em}.}$
   
   
   
   
2. ${\displaystyle \sqrt[3]{343}}$$\hspace{0.5em}=\hspace{0.5em}$ .
   Solution
   
   
   ${\displaystyle \sqrt[3]{343}=\sqrt[3]{7^3}=7\hspace{0.5em}.}$
   
   
   
   

Simplify the following expressions and write each as a reduced fraction containing no powers:

1. ${\displaystyle \frac{3^3·6^3}{9·2^3·4^3}}$$\hspace{0.5em}=\hspace{0.5em}$ .
   Solution
   
   
   ${\displaystyle \frac{3^3·6^3}{9·2^3·4^3}=\frac{3^4·2^3}{2^3·4^3}=\frac{3^4}{4^3}=\frac{81}{64}\hspace{0.5em}.}$
   
   
   
   
2. ${\displaystyle 3^2·9^{-3}·{27}^6·{27}^{-2}}$$\hspace{0.5em}=\hspace{0.5em}$ .
   Solution
   
   
   ${\displaystyle 3^2·9^{-3}·{27}^{-2}·{27}^6=3^{2+2·(-3)+3·6-6}=3^8\hspace{0.5em}.}$