Question

Verify the property $x\times (y+z)=\left(x\times y\right)+\left(x\times z\right)$ of rational numbers by using $x=\frac{-1}{2}$, $y=\frac{2}{3}$and $z=\frac{3}{4}$.

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

$\begin{array}{rcl}x\times (y+z)& =& \frac{-1}{2}\times \left(\frac{2}{3}+\frac{3}{4}\right)\\ & =& \frac{-1}{2}\times \left(\frac{8+9}{12}\right)\\ & =& \frac{-1}{2}\times \frac{17}{12}\\ & =& \frac{-17}{24}\end{array}$.

Taking R.H.S , we get,

$\begin{array}{rcl}(x\times y)+(x\times z)& =& \left(\frac{-1}{2}\times \frac{2}{3}\right)+\left(\frac{-1}{2}\times \frac{3}{4}\right)\\ & =& \left(\frac{-1}{3}\right)+\left(\frac{-3}{8}\right)\\ & =& \frac{-8+(-9)}{24}\\ & =& \frac{-17}{24}\end{array}$.

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

$L.H.S=R.H.S.=\frac{-17}{24}$

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