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Properties of Rational Numbers - Distributivity of Multiplication Over Addition for Rational

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Distributive for multiplication over Addition : 
To understand this, consider the rational numbers `-3/4 , 2/3` and `-5/6`

`-3/4 xx {2/3 + (-5/6)} = -3/4 xx {((4) + (-5))/6}`

`= -3/4 xx (-1/6) = 3/24 = 1/8`

Also `-3/4 xx 2/3 = (-3 xx 2 )/(4 xx 3) = -6/12 = -1/2`

And `-3/4 xx -5/6 = 5/8`

Therefore `(-3/4 xx 2/3) + (-3/4 xx -5/6) = -1/2 + 5/8 = 1/8`

Thus, `-3/4 xx {2/3 + -5/6} = (-3/4 xx 2/3) + (-3/4 xx -5/6)` 

 

Distributive for multiplication over substraction : 

consider the rational numbers  `3/8 , -2/5 , 6/7`

`3/8 xx (-2/5 -  6/7 )` = `3/8 xx (-14 - 30)/35`

`= 3/8 xx -44/35`

`= -33/70` 

Also , `3/8 xx -2/5 - 3/8 xx 6/7`

=`-6/40 - 18 / 56` 
`= -3/20 - 9/28 = -33/70`

Distributive for Divisoin over Addition :

consider the rational numbers  are `3/8 ,-2/5 ,6/7`

a ÷ (b + c) = `3/8 ÷ (-2/5 + 6/7) =  3/8 ÷ -16/35`

`= 3/8 xx 35/-16 = -105/128`

a÷b + a÷ c = -1/2

So, a ÷ (b+c) not equal a ÷ b + a ÷ c 

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