O que seria a propriedade distributiva?

(Videos) The Distributive Property

by Math Antics

In these videos by mathantics, he goes over what the distributive property is and how you use it.

$\large{\text{Summary}}$

To distribute something means we give it to each member of a group. So, the distributive property allows us to distribute a term to a group of terms. It is also sometimes called the distributive property of multiplication.

Let's look at the example: $3\times(4+6)$. There are two ways we can simplify this. The first way is to use order of operations. The second way, we'll use the distributive property.

Order of Operations

Using this, we do the parentheses first. Then, we multiply. $$\begin{align} &3\times(4+6)\\ &=3\times(10)\\ &=\boxed{30} \end{align}$$

Distributive Property

With this property, we're going to distribute the $3$ being multiplied to each term in the parentheses. $$\begin{align} &3\times(4+6)\\ &=3\times4+3\times6\\ &=12+18\\ &=\boxed{30} \end{align}$$

Why?

If they both give the same answer, why do we use the second method? In some cases, distributing will be more work. Take the example we just did; it's much easier to add inside the parentheses to get a $10$ and then multiply.

What about this problem? $$8\times(50+3)$$ If we add first, this problem is much harder to do. If we distribute, it's easy. $$\begin{align} &8\times(50+3)\\ &=8\times50+8\times3\\ &=400+24\\ &=\boxed{424} \end{align}$$

Turning a Problem into a Distributive Problem

We can even turn problems into distributive problems to make them easier. Suppose we want to multiply: $$5\times47$$ This isn't easy to do in your head. But, suppose we wrote $47$ as $40+7$. $$5\times(40+7)$$ This is much easier to do if we distribute. $$\begin{align} &5\times(40+7)\\ &=5\times40+5\times7\\ &=200+35\\ &=\boxed{235} \end{align}$$

Subtraction

Guess what! We can also distribute into terms being subtracted. $$7\times(10-4)$$ We distribute the multiplication just like before, only we subtract the terms instead of add them. $$\begin{align} &7\times(10-4)\\ &=7\times10-7\times4\\ &=70-28\\ &=\boxed{42} \end{align}$$

Mix and Match

We can even distribute if there's a mix of addition and subtraction in the group. $$\begin{align} &4\times(20+5-8)\\ &=4\times20+4\times5-4\times8\\ &=80+20-32\\ &=\boxed{68} \end{align}$$

Doesn't Work

Distributing doesn't work when the members of the group are being multiplied or divided. $$5\times(2\times3)\quad{\large{\ne}}\quad5\times2\times5\times3\\ 4\times(6\div2)\quad{\large{\ne}}\quad4\times6\div4\times2$$

In other words, we can't use multiplication and division inside the parentheses.

Variables

We can distribute if the group contains variables. $$3\times(x+6)$$ We distribute the $3$ to each term regardless of if it's a variable or not. $$\begin{align} &3\times(x+6)\\ &=3\times x+3\times6\\ &=3x+18 \end{align}$$

Here's an example with only variables. $$\begin{align} &a\times(b-c)\\ &=a\times b-a\times c\\ &=ab-ac \end{align}$$

$$\textbf{Distributive Property}$$

Below is a visual representation of the distributive property. Imagine two rectangles with the same height but different width, there are $2$ ways to calculate the total area of the two rectangles.

The Distributive Property

The distributive property says that if you're multiplying an expression in parentheses by a number, you have to do that operation to every number within the parentheses. When you know what all the numbers are, that's easy: you can just follow the order of operations and add/subtract the numbers within the parentheses first.

If we follow the order of operations, we solve this by going through the following steps: $$\begin{align} &4 \times (3 + 5)\\ &= 4 \times (8)\\ &= 32 \end{align}$$

But because of the distributive property, we'll get the same answer if we just multiply both numbers within the parentheses by $4$ and then add them together: $$\begin{align} &4 \times (3 + 5)\\ &=(4\times3) + (4\times5)\\ &= 12 + 20\\ &= 32 \end{align}$$

But what if we don't know what one (or more) of the numbers within the parentheses is?

Applying the distributive property to variables

If one of the numbers within the parentheses is unknown (a variable), we can just use the second method above (multiplying both numbers by the number outside the parentheses, then adding/subtracting them ).

Let's try it with the expression $6 \times (x - 3)$.

$$6 \times (x - 3) = 6x - 18$$

To figure this out, you have to multiply all the terms within the parentheses by $6$ (and notice that the operation, the $-$, stays the same).

But what if there are more than two terms in the parentheses?

The distributive property still applies! Just multiply each term inside the parentheses by the number immediately outside the parentheses, then add or subtract them as you would have anyway.

$$2 \times (3x - 1 + 4y) = 6x - 2 + 8y$$

We can get this by multiplying $2$ by each of the numbers inside the parentheses, then making sure we're still doing all the same operations to them. Also, notice that the number outside the parentheses can still be multiplied by the whole number attached to each variable.