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>.