Question

Verify the property $x\times (y\times z)=(x\times y)\times z$ of rational numbers by using
(a)$x=1,y=\frac{-1}{2},z=\frac{1}{4}$
(b)$x=\frac{2}{3},y=\frac{-3}{7},z=\frac{1}{2}$

Answer 1

(a) $x\times \left(y\times z\right)$

$=1\times \left(\frac{-1}{2}\times \frac{1}{4}\right)$

$=1\times \left(\frac{-1}{8}\right)$

$=\frac{-1}{8}$

$\left(x\times y\right)\times z$

$=\left(1\times \frac{-1}{2}\right)\times \frac{1}{4}$

$=\frac{-1}{2}\times \frac{1}{4}$

$=\frac{-1}{8}$

So $x\times \left(y\times z\right)=\left(x\times y\right)z$

$\left(b\right)x\times \left(y\times z\right)=\frac{2}{3}\times \left(\frac{-3}{7}\times \frac{1}{2}\right)$

$=\frac{2}{3}\times \frac{-3}{14}$

$=\frac{-6}{42}$

$\left(x\times y\right)\times z=\left(\frac{2}{3}\times \frac{-3}{7}\right)$

$=\frac{-6}{21}\times \frac{1}{2}=\frac{-6}{42}$

$=\frac{-6}{2}$

So, $x\times \left(y+z\right)=\left(x\times y\right)\times z$