Chapter 1 - XYZ Custom Plus

48Chapter 1 The BasicsAssociative Property of AdditionIn symbols: a + (b + c) = (a + b) + cIn words:Changing the grouping of the numbers in a sum will not changethe result.Associative Property of MultiplicationIn symbols: a(bc) = (ab)cIn words:Changing the grouping of the numbers in a product will not changethe result.The following examples illustrate how the associative properties can be used tosimplify expressions that involve both numbers and variables.Simplify.4. 3 + (7 + x)Example 4Simplify 4 + (5 + x).Solution 4 + (5 + x) = (4 + 5) + x Associative property of addition= 9 + x Addition5. 3(4x)Example 5Simplify 5(2x).Solution 5(2x) = (5 ⋅ 2)x Associative property of multiplication= 10x Multiplication6. ​_1 ​( 3 x)3Example 6Simplify ​_1 ​( 5 x).5Solution​_1 5 ​( 5 x) = ​ _ ​ 1 5 ​⋅ 5 ​x Associative property of multiplication= 1x Multiplication= x7. 4 ​ ​ 1 _4 ​x ​Example 7Simplify 3 ​ ​ 1 _3 ​x ​.Solution3 _​ ​ 1 3 ​x ​= ​ 3 ⋅ ​_1 3 ​ ​x Associative property of multiplication= 1x Multiplication= x8. 15 ​ ​ 3 _5 ​x ​Example 8Simplify 12 ​ ​ 2 _3 ​x ​.Solution12 ​ ​ 2 _3 ​x ​= ​ 12 ⋅ ​ 2 _3 ​ ​x Associative property of multiplication= 8x MultiplicationAnswers4. 10 + x 5. 12x 6. x 7. x8. 9xB The Distributive PropertyThe associative and commutative properties apply to problems that are either allmultiplication or all addition. There is a third basic property that involves bothaddition and multiplication. It is called the distributive property and looks like this.