Prove that |ab|=|a||b|.

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Lesson Parent: 
Rules of Absolute Values
Problem: 

Prove that |ab|=|a||b|.

Answer: 

It is true that |ab|=|a||b|.

SAMPLE SOLUTION

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Solution: 

First let's define A and B as two numbers that are greater than zero. Therefore they are always positive and A=|A| and B=|B|. In this problem we will examine cases where <Sign in to see all the formulas> and <Sign in to see all the formulas>.

To prove this we need to look at four cases:

Case a b
1 a=+A b=+B
2 a=+A b=−B
3 a=−A b=+B
4 a=−A b=−B

Now let's prove it for each case:

Case ab |ab| |a| |b| <Sign in to see all the formulas> Is <Sign in to see all the formulas>?
1 AB AB A B AB Yes
2 −AB AB A B AB Yes
3 −AB AB A B AB Yes
4 AB AB A B AB Yes

Therefore, no matter what the sign of a or b we can always say that <Sign in to see all the formulas>.