The expressions that show rational numbers are associative under multiplication follow the associative property formula:
(A×B)×C=A×(B×C)(A\times B)\times C=A\times (B\times C)(A×B)×C=A×(B×C)
where AAA, BBB, and CCC are any rational numbers. For example, if A=23A=\frac{2}{3}A=32, B=−67B=-\frac{6}{7}B=−76, and C=35C=\frac{3}{5}C=53, the associative property is demonstrated by the equality:
(23×(−67×35))=((23×−67)×35)\left(\frac{2}{3}\times \left(-\frac{6}{7}\times \frac{3}{5}\right)\right)=\left(\left(\frac{2}{3}\times -\frac{6}{7}\right)\times \frac{3}{5}\right)(32×(−76×53))=((32×−76)×53)
This shows that the way in which the rational numbers are grouped during multiplication does not affect the product
. Another example with different rational numbers 83,76,54\frac{8}{3},\frac{7}{6},\frac{5}{4}38,67,45 confirms this:
(83×76)×54=83×(76×54)=359\left(\frac{8}{3}\times \frac{7}{6}\right)\times \frac{5}{4}=\frac{8}{3}\times \left(\frac{7}{6}\times \frac{5}{4}\right)=\frac{35}{9}(38×67)×45=38×(67×45)=935
demonstrating associativity of multiplication for rational numbers
. In summary, any expression of the form
(A×B)×C=A×(B×C)(A\times B)\times C=A\times (B\times C)(A×B)×C=A×(B×C)
with rational numbers AAA, BBB, and CCC correctly shows that rational numbers are associative under multiplication.
