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Mathematics

Which of the following expressions shows that rational numbers are associative under multiplication?

  1. 45×(−67×89)=(45×−67)×89\dfrac{4}{5} \times \left(\dfrac{-6}{7} \times \dfrac{8}{9}\right) = \left(\dfrac{4}{5} \times \dfrac{-6}{7}\right) \times \dfrac{8}{9}

  2. 45×(−67×89)=45×(89×−67)\dfrac{4}{5} \times \left(\dfrac{-6}{7} \times \dfrac{8}{9}\right) = \dfrac{4}{5} \times \left(\dfrac{8}{9} \times \dfrac{-6}{7}\right)

  3. 45×(67×89)=(89×45)×−67\dfrac{4}{5} \times \left(\dfrac{6}{7} \times \dfrac{8}{9}\right) = \left(\dfrac{8}{9} \times \dfrac{4}{5}\right) \times \dfrac{-6}{7}

  4. (45×−67)×89=(−67×45)×89\left(\dfrac{4}{5} \times \dfrac{-6}{7}\right) \times \dfrac{8}{9} = \left(\dfrac{-6}{7} \times \dfrac{4}{5}\right) \times \dfrac{8}{9}

Rational Numbers

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Answer

If ab,cd and ef\dfrac{a}{b}, \dfrac{c}{d} \text{ and } \dfrac{e}{f} are any three rational numbers, then the associative law of multiplication states that ab×(cd×ef)=(ab×cd)×ef\dfrac{a}{b} \times \left(\dfrac{c}{d} \times \dfrac{e}{f}\right) = \left(\dfrac{a}{b} \times \dfrac{c}{d}\right) \times \dfrac{e}{f}

In Option 1, only the grouping of the same three rational numbers, taken in the same order, has been changed.

In Options 2 and 4, the order of the rational numbers has been changed, which shows the commutative law.

In Option 3, the order of the rational numbers has been changed and 67\dfrac{6}{7} on the left side has been replaced by −67\dfrac{-6}{7} on the right side, so the two sides are not equal at all.

Hence, Option 1 is the correct option.

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