# Use distributive property to write an equivalent expression. a. 5t + 6t + tm b. 24 + 12s + 12r

## Question:

Use distributive property to write an equivalent expression.

a. {eq}5t + 6t + tm {/eq}

b. {eq}24 + 12s + 12r {/eq}

## Applying the Distributive Property:

The distributive property is a property that states that if a, b, and c are mathematical expressions, then a(b + c) = ab + ac. Often, this property is used to multiply a mathematical expression by a sum of mathematical expressions. However, it can also be used in reverse to factor a given expression.

We can use the distributive property in reverse to rewrite each of the given expressions as an equivalent expression. The distributive property in reverse gives that if a, b, and c are mathematical expressions, then ab + ac = a(b + c). That is, if the terms of a sum of mathematical expressions share a common factor, we can pull that factor out in front of parentheses containing the sum of the remaining factors. Let's start with a.

• 5t + 6t + tm

Notice that each of the terms in this sum have a common factor of t. Therefore, we can pull t out in front of parentheses containing the sum of the remaining factors. If we factor a t out of 5t, we are left with 5. If we factor a t out of 6t, we are left with 6. If we factor a t out of tm, we are left with m. Thus, we have the following:

• 5t + 6t + tm = t(5 + 6 + m) = t(11 + m)

The distributive property gives that an equivalent expression to 5t + 6t + tm is t(11 + m).

Now, let's look at b.

• 24 + 12s + 12r

At first glance, it may not be obvious that the terms of this sum share a common factor. However, if we rewrite 24 as 12 ⋅ 2, we have the following:

• 12 ⋅ 2 + 12s + 12r

Now, it is easier to see that all three terms have a common factor of 12. Thus, we use our distributive property in reverse to pull the 12 out in front of parentheses containing the sum of the remaining factors. If we factor a 12 out of 12 ⋅ 2, we are left with 2. If we factor a 12 out of 12s, we are left with s. If we factor a 12 out of 12r, we are left with r. Thus, we have the following:

• 24 + 12s + 12r = 12 ⋅ 2 + 12s + 12r = 12(2 + s + r)

The distributive property gives that an expression that is equivalent to 24 + 12s + 12r is 12(2 + s + r). 