Question
What is the distributive property, and how can you use it to write equivalent expressions?
Answers
GPT-4o mini
The distributive property is a fundamental algebraic principle that states that for any numbers \(a\), \(b\), and \(c\):
\[
a(b + c) = ab + ac
\]
This means that when you multiply a number \(a\) by a sum \( (b + c) \), you can distribute the multiplication across each term in the sum.
Using the distributive property, you can write equivalent expressions. Here are a few examples:
1. **Example 1**:
If you have the expression \(3(x + 4)\), you can apply the distributive property:
\[
3(x + 4) = 3x + 3 \cdot 4 = 3x + 12
\]
2. **Example 2**:
For the expression \(5(2y + 3)\):
\[
5(2y + 3) = 5 \cdot 2y + 5 \cdot 3 = 10y + 15
\]
3. **Example 3**:
You can also factor an expression using the distributive property. For instance, if you have \(6x + 12\), you can factor out the common factor:
\[
6x + 12 = 6(x + 2)
\]
This shows that you can use the distributive property for both expanding and factoring expressions. Knowing how to apply this property allows for manipulation of algebraic expressions, making it easier to simplify or solve equations.
\[
a(b + c) = ab + ac
\]
This means that when you multiply a number \(a\) by a sum \( (b + c) \), you can distribute the multiplication across each term in the sum.
Using the distributive property, you can write equivalent expressions. Here are a few examples:
1. **Example 1**:
If you have the expression \(3(x + 4)\), you can apply the distributive property:
\[
3(x + 4) = 3x + 3 \cdot 4 = 3x + 12
\]
2. **Example 2**:
For the expression \(5(2y + 3)\):
\[
5(2y + 3) = 5 \cdot 2y + 5 \cdot 3 = 10y + 15
\]
3. **Example 3**:
You can also factor an expression using the distributive property. For instance, if you have \(6x + 12\), you can factor out the common factor:
\[
6x + 12 = 6(x + 2)
\]
This shows that you can use the distributive property for both expanding and factoring expressions. Knowing how to apply this property allows for manipulation of algebraic expressions, making it easier to simplify or solve equations.