Distributive Property of Multiplication

Distributive property of multiplication:

$\color{purple}{a\times\left(b+c\right)=a\times b+a\times c}$

Intuitively, we understand, that it is correct.

Indeed, multiplication is just a shorthand for addition.

If we want to multiply 4 by 5, this means that we want to use 4 in addtion five times.

We can separately add two 4's and three 4's, and then combine the result:

${\color{red}{{{4}\times{\left({2}+{3}\right)}}}}={4}\times{5}={4}+{4}+{4}+{4}+{4}={\left({4}+{4}\right)}+{\left({4}+{4}+{4}\right)}={\color{red}{{{4}\times{2}+{4}\times{3}}}}$.

Distributive property of multiplication works for negative numbers (in fact, for real numbers) as well.

Example 1. ${5}\times{\left({3}+{\left(-{4}\right)}\right)}={5}\times{3}+{5}\times{\left(-{4}\right)}=-{5}$.

Example 2. $\frac{{1}}{{2}}\times{\left(-{5}+{3}\right)}=\frac{{1}}{{2}}\times{\left(-{5}\right)}+\frac{{1}}{{2}}\times{3}=-{1}$.

We can even go in reverse direction.

Example 3. $\frac{{5}}{{8}}\times\sqrt{{{2}}}+\frac{{5}}{{8}}\times{2}=\frac{{5}}{{8}}\times{\left(\sqrt{{{2}}}+{2}\right)}$.