Q. 79 - Intermediate Algebra

Question

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

$$\begin{gathered} a.\sqrt[3]{-864} \\ b. \sqrt[4]{-256} \end{gathered}$$

Step-by-Step Solution

Answer

The simplified expressions are,

$a.-6\sqrt[3]{4}$

b. Not a real number.

Step 1 Given expressions are,

$$\begin{gathered} a.\sqrt[3]{-864} \\ b. \sqrt[4]{-256} \end{gathered}$$

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

The product property rule 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,

$\sqrt[n]{a}=\left|a\right|$

Part a Step 2 src="data:image/svg+xml;base64,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" localid="1645117111281" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/45c737e0-90e4-421b-88ee-dc382d5b2ef4.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220223%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220223T155249Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=1375f79569a0f57560b35acf7db1afdc525763e6d0054d9404789009ba6d46c8" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/45c737e0-90e4-421b-88ee-dc382d5b2ef4.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220223%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220223T140033Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=6420fe3f59422d8a1d9608f998964d8b385cdd079d8cd72670844744e29abaed" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/45c737e0-90e4-421b-88ee-dc382d5b2ef4.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220217%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220217T170008Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=35691476567f6529ffcacd2df26fccb756c615ddba3e34e2dc91d3baad45b5bd" − 864 3   .

$\sqrt[3]{-864} =\sqrt[3]{\left(-216\right).\left(4\right)}$

Now use the product property rule.

$$\begin{gathered} \sqrt[3]{-864} =\sqrt[3]{\left(-216\right)}.\sqrt[3]{\left(4\right)} \\ =\left(-6\right)\sqrt[3]{\left(4\right)} \end{gathered}$$

Part b Step 2 − 256 4 .

$\sqrt[4]{-256}=-256$

Because, $-256$ is not perfect power of the index $4$.

So, the answer is not a real number.