Name the multiplication property of rational numbers shown below:

(i) $\dfrac{3}{5} \times \dfrac{-8}{9} = \dfrac{-8}{9} \times \dfrac{3}{5}$

(ii) $\dfrac{-3}{4} \times \Big(\dfrac{5}{7} \times \dfrac{-8}{15}\Big) = \Big(\dfrac{-3}{4} \times \dfrac{5}{7}\Big) \times \dfrac{-8}{15}$

(iii) $\dfrac{4}{5} \times \Big(\dfrac{3}{-8} + \dfrac{-4}{7}\Big) = \dfrac{4}{5} \times \dfrac{3}{-8} + \dfrac{4}{5} \times \dfrac{-4}{7}$

(iv) $\dfrac{-7}{5} \times \dfrac{5}{-7} = 1$

(v) $\dfrac{8}{-9} \times 1 = 1 \times \dfrac{8}{-9} = \dfrac{8}{-9}$

(i) Commutativity property

Reason

If $\dfrac{a}{b}$ and $\dfrac{c}{d}$ are any two rational numbers, then:

$\dfrac{a}{b} \times \dfrac{c}{d} = \dfrac{c}{d} \times \dfrac{a}{b}$

(ii) Associativity property

Reason

If $\dfrac{a}{b}, \dfrac{c}{d}$ and $\dfrac{e}{f}$ are any three rational numbers, then:

$\dfrac{a}{b} \times \Big(\dfrac{c}{d} \times \dfrac{e}{f}\Big) = \Big(\dfrac{a}{b} \times \dfrac{c}{d}\Big) \times \dfrac{e}{f}$

(iii) Distributivity property

Reason

If $\dfrac{a}{b}, \dfrac{c}{d}$ and $\dfrac{e}{f}$ are any three rational numbers, then:

$\dfrac{a}{b} \times \Big(\dfrac{c}{d} + \dfrac{e}{f}\Big) = \Big(\dfrac{a}{b} \times \dfrac{c}{d}\Big) + \Big(\dfrac{a}{b} \times \dfrac{e}{f}\Big)$

(iv) Existence of inverse

Reason

The multiplicative inverse of $\dfrac{a}{b}$ = reciprocal of $\dfrac{a}{b} = \dfrac{b}{a}$.

(v) Existence of identity

Reason

For a rational number $\dfrac{a}{b}$,

$1 \times \dfrac{a}{b} = \dfrac{a}{b} \times 1 = \dfrac{a}{b}$.