Question

In the following exercises, simplify using absolute value signs as needed.

$$\begin{gathered} a.8+\sqrt{96} \\ b.\frac{8-\sqrt{80}}{4} \end{gathered}$$

Short answer

The simplified expressions are,

$$\begin{gathered} a.8+4\sqrt{6}. \\ b.2-\sqrt{5} \end{gathered}$$

Step-by-step solution

Given expressions are,

$$\begin{gathered} a.8+\sqrt{96} \\ b.\frac{8-\sqrt{80}}{4} \end{gathered}$$

The objective is, we need to find the simplified expressions for the given expressions.

The product property for the root is,

$\sqrt[n]{ab}=\sqrt[n]{a}.\sqrt[n]{b}$and $\sqrt[n]{a}.\sqrt[n]{b}=\sqrt[n]{ab}$

Also we know that,$\sqrt[n]{a^n}=a$.

Part a Step 2: localid="1645429507768" 8 + 96.

$$\begin{gathered} 8+\sqrt{96}=8+\sqrt{2\times 2\times 2\times 2\times 2\times 3} \\ =8+\sqrt{{\left(2\right)}^4\times 6} \end{gathered}$$

Now use the product property rule.

$$\begin{gathered} 8+\sqrt{96}=8+\sqrt{{\left(2\right)}^4}.\sqrt{6} \\ =8+2^2.\sqrt{6} \\ =8+4\sqrt{6} \end{gathered}$$

Part b Step 3:localid="1645429548860" 8-804

$$\begin{gathered} \frac{8-\sqrt{80}}{4}=\frac{8-\sqrt{2\times 2\times 2\times 2\times 5}}{4} \\ =\frac{8-\sqrt{{\left(4\right)}^2\times 5}}{4} \end{gathered}$$

Now use the product property rule.

$$\begin{gathered} \frac{8-\sqrt{80}}{4}=\frac{8-\sqrt{{\left(4\right)}^2}.\sqrt{5}}{4} \\ =\frac{8-4\sqrt{5}}{4} \\ =2-\sqrt{5} \end{gathered}$$