# Proving Abolutee-Value Identities: |x^2| = |x|^2, |x^n| = |x|^n, |x| = sqrt{x^2}, ...

#### char007

##### New member
Can anyone help me to prove these identities?

$$\displaystyle \mbox{(a) }\, \big|\,x^2\,\big|\, =\, \big|\,x\,\big|^2$$

$$\displaystyle \mbox{(b) }\, \big|\, x^n\, \big|\, =\, \big|\, x\, \big|^n\, \mbox{ for every integer }\, n$$

$$\displaystyle \mbox{(c) }\, \big|\, x\, \big|\, =\, \sqrt{\strut x^2\,}$$

$$\displaystyle \mbox{(d) }\, \big|\, x\, -\, y\,\big|\, \leq\, \big|\, x\, \big|\, +\, \big|\, y\, \big|$$

$$\displaystyle \mbox{(e) }\, \big|\, x\, -\, y\, \big|\, \geq\, \bigg|\big|\, x\, \big|\, -\, \big|\, y\, \big|\bigg|$$

Hint: In (e), prove that $$\displaystyle \big|\, x\, -\, y\, \big|\, \geq\, \big|\, x\, \big|\, -\, \big|\, y\, \big|$$ and $$\displaystyle \big|\, x\, -\, y\, \big|\, \geq\, \big|\, y\, \big|\, -\, \big|\, x\, \big|$$

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#### tkhunny

##### Moderator
Staff member
Can anyone help me to prove these identities?

$$\displaystyle \mbox{(a) }\, \big|\,x^2\,\big|\, =\, \big|\,x\,\big|^2$$

$$\displaystyle \mbox{(b) }\, \big|\, x^n\, \big|\, =\, \big|\, x\, \big|^n\, \mbox{ for every integer }\, n$$

$$\displaystyle \mbox{(c) }\, \big|\, x\, \big|\, =\, \sqrt{\strut x^2\,}$$

$$\displaystyle \mbox{(d) }\, \big|\, x\, -\, y\,\big|\, \leq\, \big|\, x\, \big|\, +\, \big|\, y\, \big|$$

$$\displaystyle \mbox{(e) }\, \big|\, x\, -\, y\, \big|\, \geq\, \bigg|\big|\, x\, \big|\, -\, \big|\, y\, \big|\bigg|$$

Hint: In (e), prove that $$\displaystyle \big|\, x\, -\, y\, \big|\, \geq\, \big|\, x\, \big|\, -\, \big|\, y\, \big|$$ and $$\displaystyle \big|\, x\, -\, y\, \big|\, \geq\, \big|\, y\, \big|\, -\, \big|\, x\, \big|$$

If x > 0, then |x| = x
If x < 0, then |x| = -x

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#### Jomo

##### Elite Member
Can anyone help me to prove these identities?

$$\displaystyle \mbox{(a) }\, \big|\,x^2\,\big|\, =\, \big|\,x\,\big|^2$$

$$\displaystyle \mbox{(b) }\, \big|\, x^n\, \big|\, =\, \big|\, x\, \big|^n\, \mbox{ for every integer }\, n$$

$$\displaystyle \mbox{(c) }\, \big|\, x\, \big|\, =\, \sqrt{\strut x^2\,}$$

$$\displaystyle \mbox{(d) }\, \big|\, x\, -\, y\,\big|\, \leq\, \big|\, x\, \big|\, +\, \big|\, y\, \big|$$

$$\displaystyle \mbox{(e) }\, \big|\, x\, -\, y\, \big|\, \geq\, \bigg|\big|\, x\, \big|\, -\, \big|\, y\, \big|\bigg|$$

Hint: In (e), prove that $$\displaystyle \big|\, x\, -\, y\, \big|\, \geq\, \big|\, x\, \big|\, -\, \big|\, y\, \big|$$ and $$\displaystyle \big|\, x\, -\, y\, \big|\, \geq\, \big|\, y\, \big|\, -\, \big|\, x\, \big|$$
Hi. Sure someone here can help. That is never a problem. But 1st you need to tell us where you are stuck or what you have tried and then once we see where you are going wrong then we can help. Thanks.

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#### Dr.Peterson

##### Elite Member
Can anyone help me to prove these identities?

$$\displaystyle \mbox{(a) }\, \big|\,x^2\,\big|\, =\, \big|\,x\,\big|^2$$

$$\displaystyle \mbox{(b) }\, \big|\, x^n\, \big|\, =\, \big|\, x\, \big|^n\, \mbox{ for every integer }\, n$$

$$\displaystyle \mbox{(c) }\, \big|\, x\, \big|\, =\, \sqrt{\strut x^2\,}$$

$$\displaystyle \mbox{(d) }\, \big|\, x\, -\, y\,\big|\, \leq\, \big|\, x\, \big|\, +\, \big|\, y\, \big|$$

$$\displaystyle \mbox{(e) }\, \big|\, x\, -\, y\, \big|\, \geq\, \bigg|\big|\, x\, \big|\, -\, \big|\, y\, \big|\bigg|$$

Hint: In (e), prove that $$\displaystyle \big|\, x\, -\, y\, \big|\, \geq\, \big|\, x\, \big|\, -\, \big|\, y\, \big|$$ and $$\displaystyle \big|\, x\, -\, y\, \big|\, \geq\, \big|\, y\, \big|\, -\, \big|\, x\, \big|$$
As tkhunny said, the basic way to prove each of these is to use separate cases. The first three will be similar: show that it is true for all x >= 0, and then show that it is true for all x < 0. The last two will require also considering other conditions, such as whether x-y is >=0 or <0.

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