Question

fill in the boxes to add the expressions.
(4a + 7.45)+(2.97a + 8.46) = (4a + _)+(7.45 + _)
=
(4a + 7.45)+(2.97a + 8.46) = (4a + square)+(7.45 + square)
=
square
(simplify your answer. use integers or decimals for any numbers in the expression.)

Explanation:

Step1: Apply Commutative and Associative Properties

We use the commutative and associative properties of addition to group like terms (terms with \(a\) and constant terms). For the expression \((4a + 7.45)+(2.97a + 8.46)\), we can rewrite it as \((4a + 2.97a)+(7.45 + 8.46)\) by grouping the \(a\)-terms together and the constant terms together.

Step2: Fill in the First Two Boxes

In the first box (for the \(a\)-term group), we have \(2.97a\), and in the second box (for the constant term group), we have \(8.46\).

Step3: Simplify the \(a\)-terms

Now, we simplify \(4a+2.97a\). Combining like terms, we add the coefficients: \(4 + 2.97 = 6.97\), so \(4a+2.97a = 6.97a\).

Step4: Simplify the Constant Terms

Next, we simplify \(7.45 + 8.46\). Adding these two decimals: \(7.45+8.46 = 15.91\).

Step5: Combine the Simplified Terms

Putting it all together, the simplified expression is \(6.97a+15.91\). But for the middle step where we group the terms, the first group of \(a\)-terms is \(4a + 2.97a\) and the constant group is \(7.45 + 8.46\), and then when we simplify the \(a\)-terms we get \(6.97a\) and the constants give \(15.91\), but the first part of filling the boxes: the first box (after \(4a+\)) is \(2.97a\), the second box (after \(7.45+\)) is \(8.46\), and then when we simplify the \(a\)-terms, \(4a + 2.97a=6.97a\) and the constants \(7.45 + 8.46 = 15.91\), so the final simplified expression is \(6.97a+15.91\). But focusing on the first part of filling the boxes:

First, for \((4a + 7.45)+(2.97a + 8.46)=(4a+\boldsymbol{2.97a})+(7.45+\boldsymbol{8.46})\)

Then, simplifying \(4a + 2.97a=\boldsymbol{6.97a}\) and \(7.45 + 8.46=\boldsymbol{15.91}\), so the final expression is \(6.97a + 15.91\). But the question seems to have two parts: first, filling the boxes in the grouping step, then simplifying. So:

First, the first box (the \(a\)-term to group with \(4a\)) is \(2.97a\), the second box (the constant to group with \(7.45\)) is \(8.46\). Then, when we simplify \(4a + 2.97a\), we get \(6.97a\), and \(7.45+8.46 = 15.91\), so the simplified expression is \(6.97a + 15.91\). But for the middle step of filling the boxes:

1. The first box (after \(4a+\)): \(2.97a\)
2. The second box (after \(7.45+\)): \(8.46\)
3. Then, when we simplify \(4a + 2.97a\), we get \(6.97a\) (this would be the box for the simplified \(a\)-term) and \(7.45 + 8.46 = 15.91\) (the box for the simplified constant term). But the problem's layout:

First line: \((4a + 7.45)+(2.97a + 8.46)=(4a+\boldsymbol{\_})+(7.45+\boldsymbol{\_})\)

Second line: \(=\boldsymbol{\_}\) (simplified \(a\)-term) \(+\boldsymbol{\_}\) (simplified constant term)

So filling the first two blanks (the ones in the parentheses grouping):

First blank (after \(4a+\)): \(2.97a\)

Second blank (after \(7.45+\)): \(8.46\)

Then, simplifying \(4a + 2.97a = 6.97a\) and \(7.45 + 8.46 = 15.91\), so the next line is \(= 6.97a+15.91\)

Answer:

First box (after \(4a+\)): \(2.97a\)

Second box (after \(7.45+\)): \(8.46\)

Simplified \(a\)-term: \(6.97a\)

Simplified constant term: \(15.91\)

Final simplified expression: \(6.97a + 15.91\)

(But if we consider the first part of filling the boxes as per the problem's first equation: \((4a + 7.45)+(2.97a + 8.46)=(4a+\boldsymbol{2.97a})+(7.45+\boldsymbol{8.46})\) and then \((4a + 2.97a)+(7.45 + 8.46)=\boldsymbol{6.97a}+\boldsymbol{15.91}\))