Q. 8.70 - Intermediate Algebra

Question

Simplify:

a. $\frac{u^{{\displaystyle \frac{4}{5}}}·u^{{\displaystyle \frac{-2}{5}}}}{u^{{\displaystyle \frac{-13}{5}}}}$ b. ${\left(\frac{27x^{{\displaystyle \frac{4}{5}}}{y}^{{\displaystyle \frac{1}{6}}}}{x^{{\displaystyle \frac{1}{5}}}{y}^{{\displaystyle \frac{-5}{6}}}}\right)}^{\frac{1}{3}}$

Step-by-Step Solution

Answer

Part (a). The answer is $u^3$

Part (b). The answer is $3x^{\frac{1}{5}}y$

Part (a) Step 1. Simplifying the expression

Use the Product Property in the numerator, add the exponents.

$$\begin{gathered} \frac{u^{{\displaystyle \frac{4}{5}}}·u^{{\displaystyle \frac{-2}{5}}}}{u^{{\displaystyle \frac{-13}{5}}}} \\ \frac{u^{{\displaystyle \frac{4}{5}}-{\displaystyle \frac{2}{5}}}}{u^{{\displaystyle \frac{-13}{5}}}} \\ \frac{u^{{\displaystyle \frac{2}{5}}}}{u^{{\displaystyle \frac{-13}{5}}}} \end{gathered}$$

Use the Quotient Property, subtract the

exponents,

$$\begin{gathered} \frac{u^{{\displaystyle \frac{2}{5}}}}{u^{{\displaystyle \frac{-13}{5}}}} \\ {u}^{\frac{2}{5}-\left(\frac{-13}{5}\right)} \\ {u}^{\frac{2+13}{5}} \\ {u}^{\frac{15}{5}} \\ {u}^3 \end{gathered}$$

Part (b) Step 1. Simplifying the expression

Follow the order of operations to simplify inside the parentheses first.

Use the Quotient Property, subtract the

exponents.

$$\begin{gathered} {\left(\frac{27x^{{\displaystyle \frac{4}{5}}}{y}^{{\displaystyle \frac{1}{6}}}}{x^{{\displaystyle \frac{1}{5}}}{y}^{{\displaystyle \frac{-5}{6}}}}\right)}^{\frac{1}{3}} \\ {\left(\frac{27x^{{\displaystyle \frac{4}{5}-\frac{1}{5}}}}{y^{{\displaystyle \frac{-5}{6}-\frac{1}{6}}}}\right)}^{\frac{1}{3}} \\ {\left(\frac{27x^{{\displaystyle \frac{3}{5}}}}{y^{-1}}\right)}^{\frac{1}{3}} \end{gathered}$$

Use the Product to a Power Property,

multiply the exponents.

$$\begin{gathered} {\left(\frac{27x^{{\displaystyle \frac{3}{5}}}}{y^{-1}}\right)}^{\frac{1}{3}} \\ \frac{{27}^{{\displaystyle \frac{1}{3}}}·x^{{\displaystyle \frac{3}{5}}·{\displaystyle \frac{1}{3}}}}{y^{-1}} \\ 3x^{\frac{1}{5}}y \end{gathered}$$