Sample Solution Write a mathematical proof of the algebraic equivalence of () and (). () = () associative property = () commutative property Lesson Summary Properties of Arithmetic THE COMMUTATIVE PROPERTY OF ADDITION: If and are real numbers, then + = +. THE ASSOCIATIVE PROPERTY OF ADDITION: If,, and are real numbers, then ( + ) + = +( + ). THE COMMUTATIVE PROPERTY OF MULTIPLICATION: If and are real numbers, then =. THE ASSOCIATIVE PROPERTY OF MULTIPLICATION: If,, and are real numbers, then () = (). THE DISTRIBUTIVE PROPERTY OF MULTIPLICATION: If,, and are real numbers, then a(b +) = +. Homework Problem Set 1. The following portion of a flow diagram shows that the expression + is equivalent to the expression +. Fill in each circle with the appropriate symbol: Either (for the commutative property of addition) or (for the commutative property of multiplication). Properties 103
2. Fill in the blanks of this proof showing that ( +5)( +2) is equivalent to +7 +10. Write either commutative property, associative property, or distributive property in each blank. ( + )( + ) = ( + ) +( + ) distributive property = ( + ) +( + ) commutative property = ( + ) + ( + ) commutative property = + + ( + ) distributive property = + + ( + ) commutative property = + + + distributive property = +( + )+ associative property = + + CHALLENGE PROBLEM 3. Fill in each circle of the following flow diagram with one of the letters: C for commutative property (for either addition or multiplication), A for associative property (for either addition or multiplication), or D for distributive property. Properties 104
4. What is a quick way to see that the value of the sum 53 + 18 + 47 + 82 is 200? + + + = ( + ) + ( + ) = + 5. If =37 and = 1, what is the value of the product? Give some indication as to 37 how you used the commutative and associative properties of multiplication. = ()() = by two applications of the commutative property of multiplication and by the associative property of multiplication 6. The following is a proof of the algebraic equivalency of (2) and 8. Fill in each of the blanks with either the statement commutative property or associative property. () = = ( )( ) associative property = ()() commutative property = ( ) associative property = () commutative property =()() = associative property 7. Write a mathematical proof of the algebraic equivalency of () and. () = ()() = () associative property = () commutative property = ()() associative property = 8. Write a mathematical proof to show that ( + )( + ) is equivalent to + + +. ( + )( + ) = ( + ) + ( + ) (D) = ( + ) + ( + ) (C) = + + + (D) = + + + (C) Properties 105
Spiral Review 9. Recall the following rules of exponents: = = ( ) = () = = Here,,, and are real numbers with and nonzero. Replace each of the following expressions with an equivalent expression in which the variable of the expression appears only once with a positive number for its exponent. (For example, is equivalent to.) A. (16 ) (16 ) B. (2) (2) C. (9 )(3 ) D. (25 ) (5 ) (5 ) E. (25 ) (5 ) (5 ) Properties 106
Challenge Problem 10. Grizelda has invented a new operation that she calls the average operator. For any two real numbers and, she declares to be the average of and : + = 2 a. Does the average operator satisfy a commutative property? That is, does = for all real numbers and? Explain your reasoning. Yes, use the fact that = for any real number and the commutative property. = + = ( + ) = + ( + ) = = b. Does the average operator distribute over addition? That is, does ( + ) = () + () for all real numbers,, and? Explain your reasoning. No. For instance, ( + ) = =, whereas ( ) + ( ) = + =. Properties 107