Grouping in Multiplication Problems

Grouping multiplication with parentheses

Just as with addition and subtraction, we can use extra parentheses to indicate the order in which multiplications are to be performed. For example, when you see an arithmetic problem like $2(3(5))$, you should first perform the multiplication in the parentheses to get $2(15)$, and then do the remaining multiplication to get $30$.

Do each of these multiplication problems which use parentheses. Each problem is pictured on the grid.

Each row of this table has two problems which involve multiplication, addition or subtraction, and parentheses. Use the Switch button to switch between these two problems on the grid.

Look at any row in the table above. Are the two answers in that row the same? (yes or no)

So, when you mix addition and subtraction with multiplication, the order you do them in matters. The rule to follow is:

Do multiplication before addition or subtraction, unless the addition or subtraction is inside parentheses.

Do each of these arithmetic problems involving both multiplication and addition or subtraction.

Associativity of multiplication

In each row of the table below, there are two multiplication problems with parentheses, both illustrated on the grids to the left. Calculate each of these products.

Just as with addition, it doesn’t matter where you put the parentheses when you multiply three numbers. This fact can be written using variables as:

The associative law of multiplication: for any three numbers $a$, $b$, and $c$, it is always true that
$$ (a(b))(c)=a(b(c)) $$

So we often leave out the extra set of parentheses that shows the grouping, writing things like $2(3)(4)$ instead of $(2(3))(4)$ or $2(3(4))$.

Combining multiplication and division

In this question, in order to talk about division, we will use a zoomed-in version of the grid. The solid gray lines are still a distance of 1 apart, while dashed lines indicate fractions. So, on the grid right now, the green square has height and width 1.

Evaluate each of these expressions that use both multiplication and division. Use the Regroup button to switch between the two problems in each row. If your answer is a fraction, write it in lowest terms.

Notice that the two rows in each table are equal. This is because of this rule for combining multiplication with division:

For any three numbers $a$, $b$, and $c$, it is always true that
$$ (a/c)(b) = {a(b)}/c = a(b/c) $$

Variables and associativity of multiplication

In each row of the table below, enter the values of the expressions $4(2x)$ and $(4(2))x$ for the given value of $x$. Both expressions are illustrated to the left, and controlled by the same slider.

$x$$\table 4(2x) ; (4(2))x$

Each row of this table starts with a product (multiplication) of two numbers and a variable. Using the associative law of multiplication, simplify those expressions, as shown in the example in the first row.

Variables, multiplication, and division

Each row of the tables below starts with some expression involving a variable and both multiplication and division. Simplify the expression until it involves only multiplication, as shown in the example in the first row on the left.