9.8 Rational exponents  (Page 3/7)

Solution

ⓐ
$\begin{array}{cccc}& & & \text{−}{25}^{\frac{3}{2}}\hfill \\ \\ \\ \text{Rewrite in radical form.}\hfill & & & \text{−}{\left(\sqrt{25}\right)}^3\hfill \\ \\ \\ \text{Simplify the radical.}\hfill & & & \text{−}{\left(5\right)}^3\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \mathrm{-125}\hfill \end{array}$

ⓑ
$\begin{array}{cccc}& & & \text{−}{25}^{-\frac{3}{2}}\hfill \\ \\ \\ \text{Rewrite using}{b}^{\text{−}p}=\frac{1}{b^p}.\hfill & & & \text{−}\left(\frac{1}{{25}^{\frac{3}{2}}}\right)\hfill \\ \\ \\ \text{Rewrite in radical form.}\hfill & & & \text{−}\left(\frac{1}{{\left(\sqrt{25}\right)}^3}\right)\hfill \\ \\ \\ \text{Simplify the radical.}\hfill & & & \text{−}\left(\frac{1}{{\left(5\right)}^3}\right)\hfill \\ \\ \\ \text{Simplify.}\hfill & & & -\frac{1}{125}\hfill \end{array}$

ⓒ
$\begin{array}{cccc}& & & {\left(\mathrm{-25}\right)}^{\frac{3}{2}}\hfill \\ \text{Rewrite in radical form.}\hfill & & & {\left(\sqrt{\mathrm{-25}}\right)}^3\hfill \\ \begin{array}{c}\text{There is no real number whose}\hfill \\ \text{square root is}\mathrm{-25}.\hfill \end{array}\hfill & & & \text{Not a real number.}\hfill \end{array}$

Solution

ⓐ
$\begin{array}{cccc}& & & 2^{\frac{1}{2}}·2^{\frac{5}{2}}\hfill \\ \begin{array}{c}\text{The bases are the same, so we add the}\hfill \\ \text{exponents.}\hfill \end{array}\hfill & & & 2^{\frac{1}{2}+\frac{5}{2}}\hfill \\ \text{Add the fractions.}\hfill & & & 2^{\frac{6}{2}}\hfill \\ \text{Simplify the exponent.}\hfill & & & 2^3\hfill \\ \text{Simplify.}\hfill & & & 8\hfill \end{array}$

ⓑ
$\begin{array}{cccc}& & & x^{\frac{2}{3}}·x^{\frac{4}{3}}\hfill \\ \begin{array}{c}\text{The bases are the same, so we add the}\hfill \\ \text{exponents.}\hfill \end{array}\hfill & & & x^{\frac{2}{3}+\frac{4}{3}}\hfill \\ \text{Add the fractions.}\hfill & & & x^{\frac{6}{3}}\hfill \\ \text{Simplify.}\hfill & & & x^2\hfill \end{array}$

ⓒ
$\begin{array}{cccc}& & & z^{\frac{3}{4}}·z^{\frac{5}{4}}\hfill \\ \begin{array}{c}\text{The bases are the same, so we add the}\hfill \\ \text{exponents.}\hfill \end{array}\hfill & & & z^{\frac{3}{4}+\frac{5}{4}}\hfill \\ \text{Add the fractions.}\hfill & & & z^{\frac{8}{4}}\hfill \\ \text{Simplify.}\hfill & & & z^2\hfill \end{array}$

Solution

ⓐ
$\begin{array}{cccc}& & & {\left(x^4\right)}^{\frac{1}{2}}\hfill \\ \begin{array}{c}\text{To raise a power to a power, we multiply}\hfill \\ \text{the exponents.}\hfill \end{array}\hfill & & & x^{4·\frac{1}{2}}\hfill \\ \text{Simplify.}\hfill & & & x^2\hfill \end{array}$

ⓑ
$\begin{array}{cccc}& & & {\left(y^6\right)}^{\frac{1}{3}}\hfill \\ \begin{array}{c}\text{To raise a power to a power, we multiply}\hfill \\ \text{the exponents.}\hfill \end{array}\hfill & & & y^{6·\frac{1}{3}}\hfill \\ \text{Simplify.}\hfill & & & y^2\hfill \end{array}$

ⓒ
$\begin{array}{cccc}& & & {\left(z^9\right)}^{\frac{2}{3}}\hfill \\ \begin{array}{c}\text{To raise a power to a power, we multiply}\hfill \\ \text{the exponents.}\hfill \end{array}\hfill & & & z^{9·\frac{2}{3}}\hfill \\ \text{Simplify.}\hfill & & & z^6\hfill \end{array}$

Solution

1. ⓐ
   $\begin{array}{cccc}& & & \frac{x^{\frac{4}{3}}}{x^{\frac{1}{3}}}\hfill \\ \begin{array}{c}\text{To divide with the same base, we subtract}\hfill \\ \text{the exponents.}\hfill \end{array}\hfill & & & x^{\frac{4}{3}-\frac{1}{3}}\hfill \\ \text{Simplify.}\hfill & & & x\hfill \end{array}$
   
   
2. ⓑ
   $\begin{array}{cccc}& & & \frac{y^{\frac{3}{4}}}{y^{\frac{1}{4}}}\hfill \\ \begin{array}{c}\text{To divide with the same base, we subtract}\hfill \\ \text{the exponents.}\hfill \end{array}\hfill & & & y^{\frac{3}{4}-\frac{1}{4}}\hfill \\ \text{Simplify.}\hfill & & & y^{\frac{1}{2}}\hfill \end{array}$
   
   
3. ⓒ
   $\begin{array}{cccc}& & & \frac{z^{\frac{2}{3}}}{z^{\frac{5}{3}}}\hfill \\ \begin{array}{c}\text{To divide with the same base, we subtract}\hfill \\ \text{the exponents.}\hfill \end{array}\hfill & & & z^{\frac{2}{3}-\frac{5}{3}}\hfill \\ \text{Rewrite without a negative exponent.}\hfill & & & \frac{1}{z}\hfill \end{array}$

Solution

1. ⓐ
   $\begin{array}{cccc}& & & {\left(27u^{\frac{1}{2}}\right)}^{\frac{2}{3}}\hfill \\ \\ \\ \begin{array}{c}\text{First we use the Product to a Power}\hfill \\ \text{Property.}\hfill \end{array}\hfill & & & {\left(27\right)}^{\frac{2}{3}}{\left(u^{\frac{1}{2}}\right)}^{\frac{2}{3}}\hfill \\ \\ \\ \text{Rewrite 27 as a power of 3.}\hfill & & & {\left(3^3\right)}^{\frac{2}{3}}{\left(u^{\frac{1}{2}}\right)}^{\frac{2}{3}}\hfill \\ \\ \\ \begin{array}{c}\text{To raise a power to a power, we multiply}\hfill \\ \text{the exponents.}\hfill \end{array}\hfill & & & \left(3^2\right)\left(u^{\frac{1}{3}}\right)\hfill \\ \\ \\ \text{Simplify.}\hfill & & & 9u^{\frac{1}{3}}\hfill \end{array}$
   
   
2. ⓑ
   $\begin{array}{cccc}& & & {\left(8v^{\frac{1}{4}}\right)}^{\frac{2}{3}}\hfill \\ \\ \\ \begin{array}{c}\text{First we use the Product to a Power}\hfill \\ \text{Property.}\hfill \end{array}\hfill & & & {\left(8\right)}^{\frac{2}{3}}{\left(v^{\frac{1}{4}}\right)}^{\frac{2}{3}}\hfill \\ \\ \\ \text{Rewrite 8 as a power of 2.}\hfill & & & {\left(2^3\right)}^{\frac{2}{3}}{\left(v^{\frac{1}{4}}\right)}^{\frac{2}{3}}\hfill \\ \\ \\ \begin{array}{c}\text{To raise a power to a power, we multiply}\hfill \\ \text{the exponents.}\hfill \end{array}\hfill & & & \left(2^2\right)\left(v^{\frac{1}{6}}\right)\hfill \\ \\ \\ \text{Simplify.}\hfill & & & 4v^{\frac{1}{6}}\hfill \end{array}$