Problem 25 - Beginning and Intermediate Algebra

Question

Decide whether each statement is an example of a commutative, an associative, an identity, \(a n\) inverse, or the distributive property. $$ 5 \cdot(13 \cdot 7)=(5 \cdot 13) \cdot 7 $$

Step-by-Step Solution

Answer

Associative property

Step 1: Identify the property

Examine the given math operation and identify the property being used. The equation states that multiplying 5 by the product of 13 and 7 is the same as first multiplying 5 and 13, and then multiplying the result by 7.

Step 2: Compare with properties

Recall the definitions of the different properties. The commutative property involves changing the order of operations, the associative property involves changing the grouping without changing the order, the identity property involves adding 0 or multiplying by 1, the inverse property involves combining operations that result in the identity element (0 for addition and 1 for multiplication), and the distributive property involves multiplying a number by a sum.

Step 3: Match the example

In the example, only the grouping of the numbers changes without changing the order of multiplication. This matches the associative property, which states that the way in which numbers are grouped in multiplication or addition does not affect their product or sum.

Key Concepts

Commutative Property

The commutative property tells us that the order in which we add or multiply numbers does not impact the result. For instance, consider the addition example: \(3 + 5 = 5 + 3\). No matter the order, the sum is still 8. Similarly, for multiplication: \(4 \times 6 = 6 \times 4\). Both ways, the product is 24. This property assures that switching numbers around in an addition or multiplication problem won't change the answer. It applies only to addition and multiplication, not subtraction or division.

Identity Property

The identity property involves using special 'identity' numbers that don't change the value when used in an operation. For addition, the identity number is 0 because adding 0 to any number leaves it unchanged: \(7 + 0 = 7\). For multiplication, the identity number is 1 because multiplying any number by 1 gives the original number: \(9 \times 1 = 9\). This property helps maintain the original value in equations.

Inverse Property

The inverse property involves pairing numbers with their opposites to get the identity element. In addition, the inverse of a number is its negative, because adding them results in zero: \(10 + (-10) = 0\). For multiplication, the inverse is the reciprocal, as multiplying them results in one: \(8 \times \frac{1}{8} = 1\). This property is useful in solving equations, as it helps in canceling out terms.

Distributive Property

The distributive property connects multiplication and addition. It states that multiplying a number by a sum is the same as doing each multiplication separately: \(a \times (b + c) = a \times b + a \times c\). For example, \(2 \times (3 + 4) = 2 \times 3 + 2 \times 4\). This simplifies to \(14 = 6 + 8\). This property helps break down complex problems into simpler steps.