Question

Find the product, using suitable properties: $625\times \left(-35\right)+\left(-625\right)\times 65$

Answer 1

Simplify using the distributive law of multiplication:

According to the distributive law of multiplication over addition,

$a\times b+a\times c=a\times \left(b+c\right)$

The given expression is $625\times \left(-35\right)+\left(-625\right)\times 65$.

Then, the required product is,

$625\times \left(-35\right)+\left(-625\right)\times 65=625\times \left(-35\right)+625\times \left(-65\right)$ $\left[\because \left(-a\right)\times b=a\times \left(-b\right)\right]$

$\Rightarrow 625\times \left(-35\right)+\left(-625\right)\times 65=625\times \left[\left(-35\right)+\left(-65\right)\right]$ $[\because a\times b+a\times c=a\times \left(b+c\right)]$

$\Rightarrow 625\times \left(-35\right)+\left(-625\right)\times 65=625\times \left[-35-65\right]$ $[\because +\left(-a\right)=-a]$

$\Rightarrow 625\times \left(-35\right)+\left(-625\right)\times 65=625\times \left[-100\right]$

$\Rightarrow 625\times \left(-35\right)+\left(-625\right)\times 65=-62500$

Hence, the product of $625\times \left(-35\right)+\left(-625\right)\times 65$ is $-62500$.