Q. 80 - Intermediate Algebra

Question

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

$$\begin{gathered} a. \sqrt[5]{-486} \\ b. \sqrt[6]{-64} \end{gathered}$$

Step-by-Step Solution

Answer

The simplified expressions are,

$a. -3\sqrt[5]{2}$

b. Not a real number.

Step 1 Given expressions are,

$$\begin{gathered} a. \sqrt[5]{-486} \\ b. \sqrt[6]{-64} \end{gathered}$$

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

The product property rule 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, $\sqrt[n]{a}=\left|a\right|$.

Part a Step 2 src="data:image/svg+xml;base64,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" localid="1645117344697" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/3dafdfb3-4fb3-44ca-9f00-9189763835e4.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220223%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220223T155311Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=367e728bd53cdda7109120cae586b51e53f51444ea1640aee89bec7b68aa75e0" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/3dafdfb3-4fb3-44ca-9f00-9189763835e4.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220223%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220223T141030Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=3528cef67918c77f927d05343387ea62712505fac1238170aa8feb975f10edce" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/3dafdfb3-4fb3-44ca-9f00-9189763835e4.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220217%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220217T170354Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=cb8bd312e4d404c6cc6cb8ddc09dddfd8d502ca5ff73891f311bb8c7150c6692" − 486 5 .

$$\begin{gathered} \sqrt[5]{-486}=\sqrt[5]{\left(-243\right).2} \\ =\sqrt[5]{{\left(-3\right)}^5.2} \end{gathered}$$

Now use the product property rule.

$$\begin{gathered} \sqrt[5]{-486}=\sqrt[5]{{\left(-3\right)}^5}.\sqrt[5]{2} \\ =\left(-3\right)\sqrt[5]{2} \end{gathered}$$

Part b Step 2 - 64 6 .

$\sqrt[6]{-64}=-64$

Because, there is not perfect power of the index $n$ where $n^6$ is the no real number $-64$.

So, the answer is not a real number.