How do you use mental math to find the sum or product and then identify the property or properties that you use to simplify $\left( {27 + 8} \right) + 12$?

Hint: In this question, we want to simplify the given expression using the sum or product property.
There are four main number properties to find the sum:
If ‘a’, ‘b’, and ‘c’ are numbers then,
Commutative property: $a + b = b + a$
Associative property: $\left( {a + b} \right) + c = a + \left( {b + c} \right)$
Identity property: $a + 0 = a$
Distributive property: $a \times \left( {b + c} \right) = a \times b + a \times c$
There are four main number properties to find the product:
If ‘a’, ‘b’, and ‘c’ are numbers then,
Commutative property: $a \times b = b \times a$
Associative property: $\left( {a \times b} \right) \times c = a \times \left( {b \times c} \right)$
Identity property: $a \times 1 = a$
Distributive property: $a + \left( {b \times c} \right) = \left( {a + b} \right) \times \left( {a + c} \right)$

Complete step by step solution:
In this question, the given expression is:
$ \Rightarrow \left( {27 + 8} \right) + 12$
First, let us add 27 and 8. The answer will be 35.
$ \Rightarrow 35 + 12$
Let us add 35 and 12. The answer will be 47.
$ \Rightarrow 47$
Now, let us regroup the given expression. For regrouping, we will first add 8 and 12. And then we will add 27.
That is equal to,
$ \Rightarrow 27 + \left( {8 + 12} \right)$
First, let us add 8 and 12. The answer will be 20.
$ \Rightarrow 27 + 20$
Let us add 27 and 20. The answer will be 47.
$ \Rightarrow 47$
Here, we applied the regrouping method. It uses the associative property. The associative property gets its name from the word “Associate” and it refers to the grouping of numbers. This property states that when three or more numbers are added or multiplied the sum or product is the same regardless of the grouping of the addends or multiplicands.

Note:
The additive identity property is also called the zero property of the addition.
Addition of two whole numbers except for zero will always give a bigger number.
When we add numbers except 0 on a number line, the result will always shift to the right.