Q. 8.37 - Intermediate Algebra

Question

Simplify:

$\left(a\right) 5+\sqrt{75} \left(b\right) \frac{10-\sqrt{75}}{5}.$

Step-by-Step Solution

Answer

Part a. After simplifying we get, $5(1+\sqrt{3}).$

Part b. After simplifying we get, $2-\sqrt{3}.$

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

Accordingly, we can write as

$$\begin{gathered} 5+\sqrt{75} \\ =5+\sqrt{25\times 3} \end{gathered}$$

Part a Step 2. Rewrite the radical as the product of two radicals

Thus, $$\begin{gathered} 5+\sqrt{25·3} \\ 5+\sqrt{25}·\sqrt{3} \end{gathered}$$

Part a Step 3. Simplify

After simplifying we get,

$$\begin{gathered} 5+5\sqrt{3} \\ =5(1+\sqrt{3}). \end{gathered}$$

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

Accordingly, we can write as

$$\begin{gathered} \frac{10-\sqrt{75}}{5} \\ =\frac{10-\sqrt{25·3}}{5} \end{gathered}$$

Part b Step 2. Rewrite the radical as the product of two radicals

Thus, $$\begin{gathered} \frac{10-\sqrt{25·3}}{5} \\ \frac{10-\sqrt{25}·\sqrt{3}}{5} \end{gathered}$$

Part b Step 3. Simplify

After simplifying we get,

$$\begin{gathered} \frac{10-\sqrt{25}·\sqrt{3}}{5} \\ =\frac{10-5\sqrt{3}}{5} \\ =\frac{5(2-\sqrt{3})}{5} \\ =2-\sqrt{3} \end{gathered}$$