Why Can the Denominator of a Rational Expression Never Be Equal to Zero?

Having a denominator of zero is the same thing as dividing by zero, which is a nonsensical operation. Dividing by zero would be like the real-world equivalent of trying to share eight apples among zero people. In mathematics, the quotient of such an operation is undefined.

For example, if dividing six by zero produced a real quotient, then it would follow that multiplying that quotient by zero would give a product of six. Since any number multiplied by zero equals zero, there is no such quotient. Algebraic manipulations involving fractions with zero denominators often lead to meaningless solutions because dividing by zero is an invalid operation.