# How do you use the distributive law to multiply 12 xx 12 ?

Jan 6, 2017

See explanation...

#### Explanation:

Multiplication is distributive over addition. That is:

For any numbers $a$, $b$ and $c$:

$a \left(b + c\right) = a b + a c \text{ }$ (left distributive)

$\left(a + b\right) c = a c + b c \text{ }$ (right distributive)

So we can split the multiplication of $12 \times 12$ down like this:

$12 \times 12 = 12 \left(10 + 2\right) = \left(12 \times 10\right) + \left(12 \times 2\right) = 120 + 24 = 144$

In fact, we are using this distributive property when we use long multiplication:

$\textcolor{w h i t e}{0 + 0} 12 \textcolor{l i g h t g r a y}{0} \text{ } \leftarrow 12 \times 1 \textcolor{g r e y}{\times 10}$
$\textcolor{w h i t e}{0} + \textcolor{w h i t e}{0} \underline{\textcolor{w h i t e}{0} 24} \text{ } \leftarrow 12 \times 2 \textcolor{g r e y}{\times 1}$
$\textcolor{w h i t e}{0 + 0} 144$

$\textcolor{w h i t e}{}$

If you are only comfortable with multiplying numbers up to $10 \times 10$ then we could split it down even further, like this:

$12 \times 12 = 12 \left(10 + 2\right)$

$\textcolor{w h i t e}{12 \times 12} = \left(12 \times 10\right) + \left(12 \times 2\right)$

$\textcolor{w h i t e}{12 \times 12} = \left(\left(10 + 2\right) \times 10\right) + \left(\left(10 + 2\right) \times 2\right)$

$\textcolor{w h i t e}{12 \times 12} = \left(\left(10 \times 10\right) + \left(2 \times 10\right)\right) + \left(\left(10 \times 2\right) + \left(2 \times 2\right)\right)$

$\textcolor{w h i t e}{12 \times 12} = \left(100 + 20\right) + \left(20 + 4\right)$

$\textcolor{w h i t e}{12 \times 12} = 100 + \left(20 + 20\right) + 4$

$\textcolor{w h i t e}{12 \times 12} = 100 + 40 + 4$

$\textcolor{w h i t e}{12 \times 12} = 144$