The associative property of addition states that for any three numbers [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex], the equation [tex]\((a + b) + c = a + (b + c)\)[/tex] holds true. This property essentially allows us to regroup the terms in an addition expression without changing the result.
To determine if using the associative property of addition is a good strategy for simplifying the expression [tex]\(\left(\frac{4}{3}+\frac{2}{3}\right) + 2\)[/tex], let's break down the expression step by step.
1. Simplifying Inside the Parentheses:
[tex]\[
\left(\frac{4}{3}+\frac{2}{3}\right)
\][/tex]
Since the denominators are the same, we can add the numerators:
[tex]\[
\frac{4+2}{3} = \frac{6}{3} = 2
\][/tex]
2. Adding the Simplified Result to 2:
[tex]\[
2 + 2 = 4
\][/tex]
Now, we have the expression fully simplified to [tex]\(4\)[/tex].
To illustrate the associative property, let's approach this problem by regrouping the terms.
- Original expression:
[tex]\[
\left(\frac{4}{3}+\frac{2}{3}\right) + 2
\][/tex]
- Using the associative property to regroup:
[tex]\[
\frac{4}{3} + \left(\frac{2}{3} + 2\right)
\][/tex]
3. Simplifying Inside the New Parentheses:
[tex]\[
\frac{2}{3} + 2
\][/tex]
We first convert 2 to a fraction with the same denominator:
[tex]\[
2 = \frac{6}{3}
\][/tex]
Now, add the fractions:
[tex]\[
\frac{2}{3} + \frac{6}{3} = \frac{2+6}{3} = \frac{8}{3}
\][/tex]
4. Adding the Remaining Term:
[tex]\[
\frac{4}{3} + \frac{8}{3}
\][/tex]
Add the fractions:
[tex]\[
\frac{4+8}{3} = \frac{12}{3} = 4
\][/tex]
Both methods yield the same result, [tex]\(4\)[/tex].
### Conclusion:
While the associative property of addition is mathematically valid and can be used to regroup terms, in this specific expression, it does not necessarily simplify the process further because the expression [tex]\(\left(\frac{4}{3}+\frac{2}{3}\right) +2\)[/tex] is already straightforward to simplify directly. The property is more useful in complex expressions where regrouping significantly simplifies calculations. In this instance, applying the associative property doesn't provide additional simplification, though it confirms that the result remains consistent regardless of the grouping.
