x×y×x×y

x×y×x×y denotes the product of four factors in which x and y are multiplied in alternating order: x times y, then that result times x, then that result times y. In standard algebra with associative multiplication, this four-fold product is unambiguous and equals (xy)², the square of the product xy.

If x and y commute (that is, xy = yx), then (xy)² equals x²y², so x×y×x×y = x²y² as in ordinary arithmetic. In contrast, in noncommutative settings such as matrix multiplication or certain algebras, xy and yx need not be equal, so x×y×x×y remains xyxy and equals (xy)², but this need not be the same as x²y².

A concrete example in real numbers confirms the straightforward case: with x = 2 and y = 3, x×y×x×y = 2×3×2×3 = 36, which equals (xy)² = (6)² = 36, and also equals x²y² = 4×9 = 36 because multiplication is commutative for real numbers.

In a noncommutative setting, the distinction matters: xyxy is not generally equal to xxyy, so (xy)² need not equal x²y². The expression x×y×x×y is a standard representation of the square of the product xy, and it generalizes to higher powers (xy)ⁿ by repeating the factor xy n times. This appears in various areas of algebra, including ring theory and group theory, where the order of factors affects the outcome unless commutativity holds.