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Identify the properties used for the first two steps in simplifying \(6ab^2 + 4a^2b - 3ab^2 + 3a^2b - 2a^2b^2\).

[tex]\[
\begin{aligned}
6ab^2 + 4a^2b - 3ab^2 + 3a^2b - 2a^2b^2 & = 4a^2b + 3a^2b - 3ab^2 + 6ab^2 - 2a^2b^2 \\
& = (4 + 3)a^2b + (-3 + 6)ab^2 - 2a^2b^2 \\
& = 7a^2b + 3ab^2 - 2a^2b^2
\end{aligned}
\][/tex]

What properties are used for the first two steps in this simplification?

Sagot :

GinnyAnswer
To simplify the expression \(6ab^2 + 4a^2b - 3ab^2 + 3a^2b - 2a^2b^2\), we utilize a couple of algebraic properties. Here's a detailed, step-by-step explanation:

### Step 1: Use the Commutative Property of Addition to Rearrange the Terms

The Commutative Property of Addition states that you can add terms in any order. This property allows us to rearrange the terms in the given expression to group like terms together.

Given expression:
[tex]\[6ab^2 + 4a^2b - 3ab^2 + 3a^2b - 2a^2b^2\][/tex]

Rearrange the terms to group like terms together:
[tex]\[4a^2b + 3a^2b - 3ab^2 + 6ab^2 - 2a^2b^2\][/tex]

### Step 2: Use the Associative Property of Addition to Group Like Terms

The Associative Property of Addition states that you can group terms in any way without changing the sum. This allows us to combine the coefficients of like terms.

Rearranged expression:
[tex]\[4a^2b + 3a^2b - 3ab^2 + 6ab^2 - 2a^2b^2\][/tex]

Group the like terms:
[tex]\[(4 + 3)a^2b + (-3 + 6)ab^2 - 2a^2b^2\][/tex]

Combining these yields:
[tex]\[7a^2b + 3ab^2 - 2a^2b^2\][/tex]

Thus, the identified properties used in the first two steps are the Commutative Property of Addition and the Associative Property of Addition.
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