Question

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

$$\begin{gathered} a.3+\sqrt{125} \\ b.\frac{15+\sqrt{75}}{5} \end{gathered}$$

Short answer

The simplified expressions are,

$$\begin{gathered} a.3+5\sqrt{5} \\ b.3+\sqrt{3} \end{gathered}$$

Step-by-step solution

Step 1 Given expressions are,

$$\begin{gathered} a.3+\sqrt{125} \\ b.\frac{15+\sqrt{75}}{5} \end{gathered}$$

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

The product property for the roots 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 2src="data:image/svg+xml;base64,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" localid="1645429398151" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/5b2f93d9-de8f-45f0-ada7-5777ed4f3765.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220223%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220223T155336Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=d0b620b309780fe5232927e2b48af9d7fe342fd03b65a36e9d7fab52d46e5a2b" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/5b2f93d9-de8f-45f0-ada7-5777ed4f3765.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220221%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220221T074435Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=58c6267f1f7bf0dafff2076bfd77fe81a86cd106d2e6581804c34327a51523f8" 3+ 125.

$$\begin{gathered} 3+\sqrt{125}=3+\sqrt{5\times 5\times 5} \\ =3+\sqrt{{\left(5\right)}^2\times 5} \end{gathered}$$

Now use the product property rule.

$$\begin{gathered} 3+\sqrt{125}=3+\sqrt{{\left(5\right)}^2}.\sqrt{5} \\ =3+5\sqrt{5} \end{gathered}$$

Part b Step 2  15 + 755.

$$\begin{gathered} \frac{15+\sqrt{75}}{5}=\frac{15+\sqrt{5\times 5\times 3}}{5} \\ =\frac{15+\sqrt{{\left(5\right)}^2\times 3}}{5} \end{gathered}$$

Now use the product property rule.

$$\begin{gathered} \frac{15+\sqrt{75}}{5}=\frac{15+\sqrt{{\left(5\right)}^2}.\sqrt{3}}{5} \\ =\frac{15+5\sqrt{3}}{5} \\ =\frac{5\left(3+\sqrt{3}\right)}{5} \\ =3+\sqrt{3} \end{gathered}$$