a×b×ca×b×c

a×b×ca×b×c is a product of five factors: a, b, ca, b, and c. In standard algebra, juxtaposition denotes multiplication, so ca is read as the product of c and a (that is, c × a). With this interpretation, the expression becomes a × b × (c × a) × b × c.

Using the associativity and commutativity of multiplication, the factors can be rearranged and grouped: a × b × (c × a) × b × c = (a × a) × (b × b) × (c × c) = a^2 × b^2 × c^2. Therefore the expression simplifies to (abc)^2.

If, instead, ca were intended as a single distinct variable named “ca,” the product would be a × b × ca × b × c, which does not simplify further without extra relations between the variables. In typical algebraic practice, however, ca is treated as c × a, and the simplification to a^2 b^2 c^2 holds.

This example illustrates how the rules of multiplication in a commutative setting allow repeated factors to be combined into powers: a twice, b twice, and c twice, yielding (abc)^2. It also highlights the importance of clear notation, since ambiguity about whether ca denotes a composite factor or a product can affect the interpretation. See also exponent rules and commutative algebra for related ideas.