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Chapter 8: Q. 83 (page 771)

In the following exercises, simplify using absolute value signs as needed. a.5+12b.10−242

Short Answer

Expert verified

The simplified expressions are,

a.5+23b.5-6

Step by step solution

01

Step 1 Given expressions are,

a.5+12b.10−242

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

The product property for the root is,

abn=an.bnand an.bn=abn.

Also we know that,an=a.

02

Part a Step 2src="data:image/svg+xml;base64,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" localid="1645429632816" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/7b8f17b4-de00-4ee2-b9ef-16fc521c3cca.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220223%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220223T155325Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=abf7ac98a62a3de4f229872bed1629ca26bb6db8b7044b752aaf22854fcc3c9d" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/7b8f17b4-de00-4ee2-b9ef-16fc521c3cca.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220221%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220221T075415Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=80942fe2f01b2d7e706d51d284036abbe943428b9f8e3b3e145eab9e756e5b76"  5 +12

5+12=5+2×2×3=5+22×3

Now use the product property rule.

localid="1644760767236" 5+12=5+22.3=5+23

03

Part b Step 2 10-242.

10−242=10-2×2×2×32=10-22×62

Now use the product property rule.

10-242=10-22.62=10-262=25-62=5-6

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