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Rule

If $ab=0,$ then $a=0$ or $b=0.$

This fact is also true if $a$ and $b$ are algebraic expressions.

Let $a$ and $b$ be two real numbers such that $ab=0.$ The goal is to show that at least one of them is zero. If $a=0$ or $b=0,$ there is nothing to prove and the property is true. The interesting case is when one of the numbers is different from zero. Therefore, assume that $a$ is a non-zero number.
It was obtained that $b=0.$ Consequently, if the product $ab$ is equal to $0$ and $a =0,$ then $b=0.$ This shows that when the product of two numbers is zero, at least one of them must be zero.

$ab=0anda =0 $

Since $a =0,$ it has a multiplicative inverse, which is $a1 .$ Next, multiply both sides of the left-hand equation by $a1 $ and simplify.
$ab=0$

MultEqn

$LHS⋅a1 =RHS⋅a1 $

$a1 ⋅ab=a1 ⋅0$

▼

Simplify

ZeroPropMult

Zero Property of Multiplication

$a1 ⋅ab=0$

MoveRightFacToNum

$ca ⋅b=ca⋅b $

$aab =0$

CrossCommonFac

Cross out common factors

$a a b =0$

CancelCommonFac

Cancel out common factors

$1b =0$

DivByOne

$1a =a$

$b=0$

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