Choose the property of real numbers that justifies each equation.
Answer (1)
The equation 1"." m = m demonstrates the Identity Property of Multiplication.
The equation 4 + ( d + 5 ) = ( 4 + d ) + 5 demonstrates the Associative Property of Addition.
The equation ( 2 + n ) "."6 = 2"."6 + n "."6 demonstrates the Distributive Property of Multiplication over Addition.
The equation b + 0 = b demonstrates the Identity Property of Addition.
Explanation
Analyzing the Equations We are given four equations and asked to identify the property of real numbers that justifies each equation. Let's analyze each equation separately.
Identity Property of Multiplication The first equation is 1"." m = m . This equation demonstrates that when any real number m is multiplied by 1, the result is the number m itself. This is the Identity Property of Multiplication .
Associative Property of Addition The second equation is 4 + ( d + 5 ) = ( 4 + d ) + 5 . This equation shows that the way we group numbers when adding them does not change the sum. In other words, whether we add d and 5 first, or 4 and d first, the result remains the same. This is the Associative Property of Addition .
Distributive Property of Multiplication over Addition The third equation is ( 2 + n ) "."6 = 2"."6 + n "."6 . This equation illustrates that multiplying a sum by a number is the same as multiplying each addend separately by the number and then adding the products. In this case, we are multiplying the sum of 2 and n by 6 , which is the same as multiplying 2 by 6 and n by 6 , and then adding the results. This is the Distributive Property of Multiplication over Addition .
Identity Property of Addition The fourth equation is b + 0 = b . This equation demonstrates that when we add 0 to any real number b , the result is the number b itself. This is the Identity Property of Addition .
Examples
Understanding the properties of real numbers is crucial in various fields, such as physics, engineering, and computer science. For example, in physics, when calculating the total force acting on an object, the associative property of addition allows us to group forces in any order without affecting the final result. Similarly, in computer graphics, the distributive property is used to efficiently apply transformations to multiple points simultaneously, optimizing rendering performance. These properties provide a foundation for simplifying complex calculations and ensuring accuracy in real-world applications.
