Q. 249 - Intermediate Algebra

Question

In the following exercise , simplify 

$$\begin{gathered} \left(i\right)\frac{\sqrt{75r^3}}{\sqrt{108r^7}} \\ \left(ii\right)\sqrt[3]{\frac{24x^7}{81x^4}} \end{gathered}$$

Step-by-Step Solution

Answer

Part (a) The solution  is $\frac{5}{6r^2}$.

Part (b) The solution  is $\frac{2x}{3}$

Step 1. Given information

The given equation are

$$\begin{gathered} \left(i\right)\frac{\sqrt{75r^3}}{\sqrt{108r^7}} \\ \left(ii\right)\sqrt[3]{\frac{24x^7}{81x^4}} \end{gathered}$$

Step 2. Calculation Part(a)

$$\begin{gathered} \left(i\right)\frac{\sqrt{75r^3}}{\sqrt{108r^7}}=\frac{\sqrt{75r^3}}{\sqrt{108r^7}} \mathrm{Write} \mathrm{the} \mathrm{expression} \\ =\sqrt{\frac{75r^3}{108r^7}} R\mathrm{ewrite} \mathrm{using} \mathrm{quotient} \mathrm{property} \\ =\sqrt{\frac{25}{36{\mathrm{r}}^4} }\mathrm{Cancel} \mathrm{out} \mathrm{common} \mathrm{factor} \\ =\frac{5}{6{\mathrm{r}}^2} \\ \mathrm{Hence} \mathrm{the} \mathrm{simplified} \mathrm{equation} \mathrm{is} \frac{5}{6{\mathrm{r}}^2}. \end{gathered}$$

Step 3. Calculation Part(b)

$$\begin{gathered} \left(ii\right)\sqrt[3]{\frac{24x^7}{81x^4}}=\sqrt[3]{\frac{24x^7}{81x^4}} \mathrm{Write} \mathrm{the} \mathrm{expression} \\ =\sqrt[3]{\frac{24{\mathrm{x}}^7}{81{\mathrm{x}}^4} } \mathrm{Rewrite} \mathrm{using} \mathrm{quotient} \mathrm{property} \\ =\sqrt[3]{\frac{8{\mathrm{x}}^3}{27}} \mathrm{Cancel} \mathrm{out} \mathrm{common} \mathrm{factors} \\ =\frac{2x}{3} \\ \mathrm{Hence} \mathrm{the} \mathrm{simplified} \mathrm{value} \mathrm{is} \frac{2x}{3} \end{gathered}$$