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Use the commutative property of multiplication to write an equivalent expression to ${53m}\text{.}$

Use the associative property of multiplication to write an equivalent expression to ${3\mathopen{}\left(5n\right)}\text{.}$

Use the commutative property of addition to write an equivalent expression to ${q+84}\text{.}$

Use the associative property of addition to write an equivalent expression to ${x+\left(20+c\right)}\text{.}$

Use the distributive property to write an equivalent expression to ${3\mathopen{}\left(r+7\right)}$ that has no grouping symbols.

Solution

To use the commutative property of multiplication, we change the order in which two factors are multiplied:
$\begin{aligned} \amp\phantom{{}={}}53m\\ \amp=m\cdot 53\text{.} \end{aligned}$

To use the associative property of multiplication, we leave factors written in their original order, but change the grouping symbols so that a different multiplication has higher priority:
$\begin{aligned} \amp\phantom{{}={}}3(5n)\\ \amp=(3\cdot5)n\text{.} \end{aligned}$
You may further simplify by carrying out the multiplication between the two numbers:
$\begin{aligned} \amp\phantom{{}={}}3(5n)\\ \amp=(3\cdot5)n\\ \amp=15n\text{.} \end{aligned}$

To use the commutative property of addition, we change the order in which two terms are added:
$\begin{aligned} \amp\phantom{{}={}}q+84\\ \amp=84+q\text{.} \end{aligned}$

To use the associative property of addition, we leave terms written in their original order, but change the grouping symbols so that a different addition has higher priority:
$\begin{aligned} \amp\phantom{{}={}}x+(20+c)\\ \amp=(x+20)+c\text{.} \end{aligned}$

To use the distributive property, we multiply the number outside the parentheses, $3\text{,}$ with each term inside the parentheses:
$\begin{aligned} \amp\phantom{{}={}}3(r+7)\\ \amp=3\cdot r+3\cdot7\\ \amp=3r+21\text{.} \end{aligned}$