https://youtu.be/tl9iten87MU

A while ago, I did a video introducing absolute values and what they were but I did not go through some very useful properties and rules of absolute values. In this video I go over these properties and show, for example, that sqrt(a<sup>2</sup>) = absolute value of a. I go over some other useful properties so make sure you watch the video!

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# Absolute Values: Properties

![Absolute Values - Properties.jpg](https://files.peakd.com/file/peakd-hive/mes/EowMx8FJjdZDnwfP2MfeB4afWmhq372gQPNiV2sqX4E8QUa7CZYYDW7SWHx19UwNc9Y.jpg)

## Recap

![image.png](https://files.peakd.com/file/peakd-hive/mes/23tHbr9ym7bm5EJRLyNxUmzJmzbxJaUnVBArup4oLewTwzRrkK4GiDwH5XbYenFM3X5GU.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/23uRLUm2MRadceS77aTP1L1eCwTFbC9PXDbMUMUg6qZgs9U4EKrQc79E9ShGR6fKUMkhw.png)

## Properties

(1). √(a<sup>2</sup>) = |a|
(2). |ab| = |a||b|
(3). |a/b| = |a|/|b| where b ≠ 0
(4). |a<sup>n</sup>| = |a|<sup>n</sup>

For the following rules, suppose a > 0, then:

(5). |x| = a if and only if x = ± a
(6). |x| < a if and only if – a < x < a 
(7). |x| > a if and only if x > a or x < – a

### Rule 1

![image.png](https://files.peakd.com/file/peakd-hive/mes/Enz44ieYkK9863yDkNjrVCrvF5CpHvyz3qUGk5Jd3GzFsUgVzez6w9RJyJ2tGHBuqZw.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/23tHdQDGmf4qiQHa7AQNUo6RT8wotubfCMxLYzYj6h4c6sZAy7NFqFwmPiS7AFJTsxMBR.png)

### Rule 2

![image.png](https://files.peakd.com/file/peakd-hive/mes/Eo4KDWWrYx6jVh9yWfB2sdfyRCxJffddfxFA6YxrvHDr4XdiDBznCJHTiBM4ux7JuKN.png)

### Rule 3

Same as Rule 2 because division can be changed into multiplication:

Let c = 1/b where b ≠ 0.


|a/b| = |ac| and this then follows the same procedure as Rule 2.

### Rule 4

(-a)<sup>n</sup> and (+a)<sup>n</sup> can only differ in sign but have the same absolute value: ±|a|<sup>n</sup>.

Thus, |a|<sup>n</sup> just means that the result is positive, n-power doesn't affect the sign, thus it is equal to |a<sup>n</sup>|.

### Rules 5, 6, 7

![image.png](https://files.peakd.com/file/peakd-hive/mes/23t8DBnQEvwsaN4o41XuWpPaXMgSeJ55NHggNnuSKfK1QFsXokhgTaUZUMV99awog5obz.png)

![image.png](https://files.peakd.com/file/peakd-hive/mes/23sxr5U7kVELHxYi6NHYTL4ZKUzzXQApSB41UzUi3PAwrfF9LJMJq6pdoAciHFHznpj7e.png)