Q. 8.40 - Intermediate Algebra

Question

Simplify:

$\left(a\right) \sqrt{\frac{98}{162}} \left(b\right) \sqrt[3]{\frac{24}{375}} \left(c\right) \sqrt[4]{\frac{4}{324}}$

Step-by-Step Solution

Answer

Part a. After simplifying we get, $\frac{7}{9}.$

Part b. After simplifying we get, $\frac{2}{5}.$

Part c. After simplifying we get, $\frac{1}{3}.$

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

So, $$\begin{gathered} \sqrt{\frac{98}{162}} \\ =\sqrt{\frac{49·2}{81·2}} \end{gathered}$$

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

Thus, $$\begin{gathered} \sqrt{\frac{49·2}{81·2}} \\ =\sqrt{\frac{49}{81}} \end{gathered}$$

Part a Step 3. Simplify

After simplifying we get, 

$$\begin{gathered} \sqrt{\frac{49}{81}} \\ =\sqrt{\frac{{\left(7\right)}^2}{{\left(9\right)}^2}} \\ =\frac{7}{9} \end{gathered}$$

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

So, $$\begin{gathered} \sqrt[3]{\frac{24}{375}} \\ =\sqrt[3]{\frac{8·3}{125·3}} \end{gathered}$$

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

Thus, $$\begin{gathered} \sqrt[3]{\frac{8·3}{125·3}} \\ =\sqrt[3]{\frac{8}{125}} \end{gathered}$$

Part b Step 3. Simplify

After simplifying we get, 

$$\begin{gathered} \sqrt[3]{\frac{8}{125}} \\ =\sqrt[3]{\frac{{\left(2\right)}^3}{{\left(5\right)}^3}} \\ =\frac{2}{5} \end{gathered}$$

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

So, $$\begin{gathered} \sqrt[4]{\frac{4}{324}} \\ =\sqrt[4]{\frac{1·4}{81·4}} \end{gathered}$$

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

Thus, $$\begin{gathered} \sqrt[4]{\frac{1·4}{81·4}} \\ \sqrt[4]{\frac{1}{81}} \end{gathered}$$

Part c Step 3. Simplify

After simplifying we get, 

$$\begin{gathered} \sqrt[4]{\frac{1}{81}} \\ =\sqrt[4]{\frac{{\left(1\right)}^4}{{\left(3\right)}^4}} \\ =\frac{1}{3} \end{gathered}$$