Question

Verify the property $x\times (y+z)=\left(x\times y\right)+\left(x\times z\right)$ of rational numbers by using $x=\frac{-2}{3}$, $y=\frac{-4}{6}$and $z=\frac{-7}{9}$.

Answer 1

Verifying the property $x\times (y+z)=\left(x\times y\right)+\left(x\times z\right)$ for given $x$, $y$ and $z$:

Taking L.H.S., we get

$\begin{array}{rcl}x\times (y+z)& =& \frac{-2}{3}\times \left(\frac{-4}{6}+\frac{(-7)}{9}\right)\\ & =& \frac{-2}{3}\times \left(\frac{-12+(-14)}{18}\right)\\ & =& \frac{-2}{3}\times \frac{(-26)}{18}\\ & =& \frac{26}{27}\end{array}$.

Taking R.H.S , we get,

$\begin{array}{rcl}(x\times y)+(x\times z)& =& \left(\frac{-2}{3}\times \frac{(-4)}{6}\right)+\left(\frac{-2}{3}\times \frac{(-7)}{9}\right)\\ & =& \left(\frac{8}{18}\right)+\left(\frac{14}{27}\right)\\ & =& \frac{4}{9}+\frac{14}{27}\\ & =& \frac{12+14}{27}\\ & =& \frac{26}{27}\end{array}$.

On comparing both L.HS. and R.H.S., we get,

$L.H.S=R.H.S.=\frac{26}{27}$

Hence, the property $x\times (y+z)=\left(x\times y\right)+\left(x\times z\right)$ is verified.