Q. 8.39 - Intermediate Algebra

Question

Simplify:

$\left(a\right) \sqrt{\frac{75}{48}} \left(b\right) \sqrt[3]{\frac{54}{250}} \left(c\right) \sqrt[4]{\frac{32}{162}}$

Step-by-Step Solution

Answer

Part a. After simplifying we get, $\frac{5}{4}.$

Part b. After simplifying we get, $\frac{3}{5}.$

Part c. After simplifying we get, $\frac{2}{3}.$

Part a Step 1. Rewrite the radicand as a product using the largest perfect square factor inside the radical

Thus, $$\begin{gathered} \sqrt{\frac{75}{48}} \\ =\sqrt{\frac{25·3}{16·3}} \end{gathered}$$

Part a Step 2. Simplify the fraction by removing common factors

So, $$\begin{gathered} \sqrt{\frac{25·3}{16·3}} \\ =\sqrt{\frac{25}{16}} \end{gathered}$$

Part a Step 3. Simplify

After simplifying we get,

$$\begin{gathered} \sqrt{\frac{25}{16}} \\ =\sqrt{\frac{{\left(5\right)}^2}{{\left(4\right)}^2}} \\ =\frac{5}{4} \end{gathered}$$

Part b Step 1. Rewrite the radicand as a product using the largest perfect square factor inside the radical

Thus, $$\begin{gathered} \sqrt[3]{\frac{54}{250}} \\ =\sqrt[3]{\frac{27·2}{125·2}} \end{gathered}$$

Part b Step 2. Simplify the fraction by removing common factors

So, $$\begin{gathered} \sqrt[3]{\frac{27·2}{125·2}} \\ =\sqrt[3]{\frac{27}{125}} \end{gathered}$$

Part b Step 3. Simplify

After simplifying we get,

$$\begin{gathered} \sqrt[3]{\frac{27}{125}} \\ =\sqrt[3]{\frac{{\left(3\right)}^3}{{\left(5\right)}^3}} \\ =\frac{3}{5} \end{gathered}$$

Part c Step 1. Rewrite the radicand as a product using the largest perfect square factor inside the radical

Thus, $$\begin{gathered} \sqrt[4]{\frac{32}{162}} \\ =\sqrt[4]{\frac{16·2}{81·2}} \end{gathered}$$

Part c Step 2. Simplify the fraction by removing common factors

So, $$\begin{gathered} \sqrt[4]{\frac{16·2}{81·2}} \\ \sqrt[4]{\frac{16}{81}} \end{gathered}$$

Part c Step 3. Simplify

After simplifying we get,

$$\begin{gathered} \sqrt[4]{\frac{16}{81}} \\ =\sqrt[4]{\frac{{\left(2\right)}^4}{{\left(3\right)}^4}} \\ =\frac{2}{3} \end{gathered}$$