Question

Verify the associative property for addition and multiplication for the rational number `(-7)/9, 5/6` and `(-4)/3`

Answer 1

Let a = `(-7)/9`, b = `5/6`, c = `(-4)/3` be the given rational numbers.

(a + b) + c = `((-7)/9 + 5/6) + ((-4)/3)`

= `((-7 xx 2 + 5 xx 3)/18) + ((-4)/3)`

= `((-14 + 15)/18) + ((-4)/3)`

= `1/18 + ((-4)/3)`

= `(1 + (-4) xx 6)/18`

= `(1 + (-24))/18`

= `(-23)/18`  ...(1)

a + (b + c) = `-7/9 + (5/6 + ((-4))/3)`

= `(-7)/9 + ((5 + (-4)2)/6)`

= `(-7)/9 + ((5 + (-8))/6)`

= `-7/9 + ((-3)/9)`

= `-7/9 + ((-1)/2)`

= `(-7 xx 2 + (-1) xx 9)/18`

= `(-14 + (-9))/18`

= `(-23)/18` ...(2)

From (1) and (2), (a + b) + c = a + (b + c) is true for rational numbers.

Given the rational number a = `(-1)/2`, b = `2/3` and c = `(-5)/6`

a × (b + c) = `(-1)/2 xx (2/3 + ((-5)/6))`

= `(-1)/2 xx (((2 xx 2) + (-5 xx 1))/6)`

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

= `(-1)/2 xx ((-1)/6)`

a × (b + c) = `1/12`  ...(1)

(a × b) + (a × c) = `((-1)/2 xx 2/3) + ((-1)/2 xx ((-5)/6))`

= `(-2)/6 + 5/12`

= `((-2 xx 2) + 5 xx 1)/12`

= `(-4 + 5)/12`

(a × b) + (a × c) = `1/12`  ...(2)

From (1) and (2) a × (b + c) = (a × b) + (a × c) is true for rational numbers.

Thus associative property.