Question

The given algebraic expression (38+83) + 38 = 38 + (83 + 38) is an example of which property?
(a) Commutative
(b) Associative
(c) Closure
(d) Distributive

Answer 1

Hint: To solve this problem, we need to know the basic properties of computing algebraic expressions. These include commutative, associative, closure and distributive properties. Then, we can get the correct answer to the above problem.


Complete step-by-step solution -

We first try to understand the basic laws (namely commutative, associative, closure and distributive properties) briefly. The commutative laws state that the order in which you add or multiply two real numbers does not affect the result. For example, 3 + 5 can be solved as 5 + 3, both will give answers as 8. Also, $3\times 4=4\times 3$ , both will give the answer as 12. Now, the associative laws state that when you add or multiply any three real numbers, the grouping (or association) of the numbers does not affect the result. For example, 3 + (4 + 5) = (3+4) +5, both will give the answer as 12. Similarly, $5\times (2\times 3)=(5\times 2)\times 3$ , both give the answer as 30. According to the distributive law, we have $a\times (b+c)=(a\times b)+(a\times c)$ for any three numbers a, b and c. For example, we would have $2\times (4+6)=(2\times 4)+(2\times 6)$ where both will evaluate to 20 as the answer. The closure law states that, for addition or multiplication, performing any arithmetic operation would yield a number which belongs to the same set. For example, addition or multiplication of two natural numbers, yield a number which would be a natural number. Clearly, in the above problem, (38+83) + 38 = 38 + (83 + 38) illustrates associative property.
Hence, the correct answer is (b) Associative.

Note: The operation of subtraction and division is not closed for natural numbers. This is because subtraction of 1 and 2, that is, 1-2 = -1 does not yield a natural number. Also, division of 1 and 2 yields 0.5, which is again not a natural number. Thus, one needs to be especially careful when applying closure law.