Question

How do you use the commutative and the associative laws to find three equivalent expressions $\left( {p + q} \right) + 60$?

Answer 1

Hint: Here, in the given question, we need to find the three equivalent expressions of $\left( {p + q} \right) + 60$, using the commutative and the associative laws. We will find the first expression using commutative law. Commutative law explains that the order of terms doesn’t matter while performing arithmetic operations. It means we can change the position or swap the numbers when adding two numbers. Now, we will find the second expression using associative law. Associative law explains that the addition of numbers is possible regardless of how they are grouped. By grouping we mean the numbers which are given inside the parenthesis (). Regardless of how numbers are parenthesized the final sum of the numbers will be the same. In the end, we will find the third expression using both commutative and associative law together.

Formula used:
The commutative law of addition states that: $x + y = y + x$
The associative law of addition states that: $\left( {x + y} \right) + z = x + \left( {y + z} \right)$

Complete step-by-step solution:
We have, $\left( {p + q} \right) + 60$
Let us first use the commutative law, $x + y = y + x$. Therefore, we get
$ \Rightarrow \left( {p + q} \right) + 60 = \left( {q + p} \right) + 60$
Now, we will use the associative law, $\left( {x + y} \right) + z = x + \left( {y + z} \right)$. Therefore, we get
$ \Rightarrow \left( {p + q} \right) + 60 = p + \left( {q + 60} \right)$
Now, we will use commutative and associative laws together. Therefore, we get
$ \Rightarrow \left( {p + q} \right) + 60 = p + \left( {60 + q} \right)$

Note: Remember that both the commutative law and the associative law works for addition and multiplication only but not for subtraction and division. Remember that commutative law holds regardless of the order of numbers while addition or multiplication. Whereas, associative law holds regardless of the grouping of numbers.