The distributive property of multiplication is one of the most used properties in mathematics. This property tells us that when we have a multiplication of the form *a*(*b*+*c*), this is equivalent to *ab*+*ac*. The distributive property helps us simplify difficult problems by allowing us to rewrite expressions.

Here, we will learn about the distributive property in detail. Then we will use some examples and practice problems to apply the concepts learned.

##### ALGEBRA

**Relevant for…**

Learning how to apply the distributive property correctly.

##### ALGEBRA

**Relevant for…**

Learning how to apply the distributive property correctly.

## Definition of the distributive property

When we have expressions of the form a (b + c), the distributive property helps us to solve them with the following process:

1. Multiply each number within the parentheses by the number outside the parentheses.

2. Add all the products.

You may wonder why we don’t follow the order of operations that tells us to evaluate what’s inside the parentheses first. The answer is that there are times when we have variables and dissimilar terms inside the parentheses.

In those cases, we cannot perform operations on those terms, so we must apply the distributive property to simplify the problems.

## Distributive property in multiplication with addition

It does not matter that we use the distributive property or follow the order of operations, we will always arrive at the same answer. In the following example, we simply follow the order of operations by simplifying what is inside the parentheses first.

$latex 5\left( {4+3} \right)=5\left( 7 \right)$

$latex =35$

Using the distributive property, we do the following:

1. We multiply or distribute the outside term to the terms inside the parentheses.

2. We combine like terms.

3. We solve the equation.

$latex 5\left( {4+3} \right)=5\left( 4 \right)+5\left( 3 \right)$

$latex =20+15$

$latex =35$

### Try solving the following exercise

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## Distributive property in multiplication with subtraction

Similar to the operation above, we perform the distributive property of multiplication with subtraction following the same rules, except that we are finding the difference instead of the sum.

$latex 7\left( {7-3} \right)=7\left( 7 \right)-7\left( 3 \right)$

$latex =49-21$

$latex =28$

Note that it does not matter if the operator is a plus or a minus. We always keep the one in parentheses.

### Try solving the following exercise

## Distributive property with variables

The distributive property is especially useful when we have to simplify equations in which we have unknown values.

Using the distributive property with variables, we can solve for *x *by following these steps:

1. Multiply or distribute the outside terms to the inside terms.

2. Combine like terms.

3. Organize the terms so that the constants and variables are on opposite sides of the equal sign.

4. Solve the equation and simplify if possible.

$latex 5\left( {x-2} \right)=20$

$latex 5\left( x \right)-5\left( 2 \right)=20$

$latex 5x-10=20$

$latex 5x-10+10=20+10$

$latex 5x=20+10$

$latex 5x=30$

$latex x=6$

### Try solving the following exercise

## Distributive property with exponents

Exponents are a notation that tell us how many times a number is multiplied by itself. When we have parentheses and exponents, we can make facilitate the simplification of expressions by using the distributive property.

1. Expand the equation

2. Multiply or distribute the first numbers of a set, the outer numbers of a set, the inner numbers of a set, and the last numbers of a set.

3. Combine like terms.

4. Solve the equation and simplify if possible.

$latex {{\left( {4x+3} \right)}^{2}}$

$latex =\left( {4x+3} \right)\left( {4x+3} \right)$

$latex =16{{x}^{2}}+12x+12x+9$

$latex =16{{x}^{2}}+24x+9$

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## Distributive property with fractions

The distributive property can also be used to simplify fractions. Solving algebraic expressions with fractions looks more difficult than it really is. With the following steps, we can facilitate this:

1. Identify the fractions and use the distributive property to eventually convert them to integers.

2. For all fractions, we find the least common multiple. That is, the smallest number that the denominators will fit into. This will allow us to add the fractions.

3. Multiply each term in the equation by the least common multiple.

4. Place the variables and the constants on opposite sides of the equal sign.

5. Combine like terms.

6. Solve the equation and simplify if possible.

$latex x+5=\frac{x}{4}+\frac{1}{2}$

$latex 4\left( {x+5} \right)=4\left( {\frac{x}{4}+\frac{1}{2}} \right)$

$latex 4x+20=x+2$

$latex 4x-x=2-20$

$latex 3x=-18$

$latex x=-6$

### Try solving the following exercise

## See also

Interested in learning more about different algebraic topics? Take a look at these pages:

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