Properties of Addition and Multiplication: Practice

The factors $a$, $b$, $c$ keep their left-to-right order, but the parentheses move from the last pair to the first.

$a \times (b \times c) = (a \times b) \times c$

A change of grouping with the order fixed is the associative property of multiplication.

Work out each side. Taking $7$ from itself leaves $0$, taking $0$ away leaves the $7$ alone, and $(7 - 3) + 3 = 4 + 3 = 7$ puts back what was taken.

$7 - 3 = 4, \qquad 3 - 7 \ne 4$

So the false statement is $7 - 3 = 3 - 7$. Reversing a subtraction swaps which number is taken from which: $3 - 7$ asks you to take $7$ away from just $3$, which drops below zero. Subtraction is not commutative.

Associativity regroups the same factors in the same order without changing any of them.

$(17 \times 4) \times 5 = 17 \times (4 \times 5)$

So the blank must be $5$.

Associativity says the choice of first pair never changes a sum.

$(a + b) + c = a + (b + c)$

Every way of filling the parentheses in gives the same value, so there is nothing left for them to say and they are dropped. A chain of subtractions is different: there the two groupings disagree, which is exactly why $10 - 4 - 2$ needs the left-to-right convention.

First the $7$ and $5$ swap places (a change of order), then $8$ and $5$ are grouped to multiply first (a change of grouping).

$8 \times 7 \times 5 \;\xrightarrow{\text{commutative}}\; 8 \times 5 \times 7 \;\xrightarrow{\text{associative}}\; (8 \times 5) \times 7 = 280$

So the steps are commutative, then associative.

A counterexample must make the two groupings disagree.

$(9 - 5) - 1 = 3, \qquad 9 - (5 - 1) = 5$

Since $3 \ne 5$, subtraction is not associative. The last option gives equal values, so it proves nothing.

Commutativity lets the two terms on the right swap places.

$41 + 23 = 23 + 41$

So the blank is $41$.

Multiplying by the identity $1$ returns the number itself, for every $a$.

$a \times 1 = a$

In contrast $a \times 0 = 0$, while $a + 1$ and $a + a = 2a$ change the value (except in special cases).

Move the $25$ next to the $4$ (commutative), then group the two pairs (associative): $4 \times 25 = 100$ and $9 \times 2 = 18$.

$4 \times 9 \times 25 \times 2 = (4 \times 25) \times (9 \times 2) = 100 \times 18 = 1800$

The three terms stay in the same order, and only the parentheses move.

$13 + (87 + 29) = (13 + 87) + 29 = 100 + 29 = 129$

Moving the parentheses is the associative property of addition. It is worth doing here, because $13 + 87$ is a round $100$ while $87 + 29$ is not.

Apply the multiplicative identity, the zero property, then the additive identity in turn.

$(46 \times 1) + (8 \times 0) + 0 = 46 + 0 + 0 = 46$

Multiplication is associative, so the grouping never matters.

$a \times (b \times c) = (a \times b) \times c$

The subtraction and division statements fail for most numbers, for instance with $a=10$, $b=4$, $c=2$.