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Algebraic Properties

1. Additive Identity Property: You can add 0 to any number, and the sum will be the same number. Example with variables: a + 0 = a  Example with numbers:  3 + 0 = 3

2. Multiplicative Identity Property: You can multiply 1 to any number, and have the product be the same sum. Example with variables: e x 1 = e Example with number: 21 x 1 = 21

3. Associative Property of Addition: You can move parentheses around any numbers and get the same sum; you can group the numbers in any way and get the same sum. Example with variables: (f + h) + k = f + ( h + k ) Example with numbers: (8 + 9) + 4 = 8 + ( 9 + 4)

4. Associative Property of Multiplication: You can move parentheses around any numbers and get the same product; you can group the numbers in any way and get the same product. Example with variables: (p x r) x a = p + (r x a)  Example with numbers: (9 x 3) x 5 = 9 + (3 x 5)

5. Commutative Property of Addition: You can add the numbers in any order and get the same sum. Example with variables: a + s + e = s + e + a Example with numbers: 6 + 8 + 2 = 8 + 2 + 6

6. Commutative Property of Multiplication: You can multiply the numbers in any order and get the same product. Example with variables: o x y x r = y x r x o Example with numbers: 9 x 3 x 7 = 3 x 7 x 9

7. Distributive Property: When you are both adding and multiplying using parentheses you can distribute the multiplication in the addition inside of the parentheses. Example with variables: j ( r + o ) = (j x r ) + ( j x o ) Example with numbers: 4 ( 5 + 2 ) = ( 4 x 5 ) + ( 4 x 2 )

8. Additive Inverse Property: You can add the inverse of any number to the number and the sum will always be 0. Example with variables: u + -u = 0 Example with numbers: 6 + -6 = 0

9. Multiplicative Inverse Property: You can multiply any number by its reciprocal and the product will always be 0. Example with variables: s/1 x 1/s = 0 Example with numbers: 4/1 x 1/4 = 0

10. Zero Product Property: If the product of two or more numbers multiplied is 0, at least one of those numbers has to be 0. Example with variables: y x i = 0, y or i = 0 Example with numbers: 3 x 0 = 0, 0 = 0

How we will use algebraic properties in real life: * If you're at the store and you buy a jug of milk for $2.75, a chocolate bar for $0.99, and a toy trucker for $7.99, it would be helpful to know that since the Associative Property states that if you add them up it does not matter the order you add them up in, because the sum will always be $17.72.
 * If you're at a potato farm, and each bag of potatoes are buy-one-get-one free at, say $6.25 a bag, then it would be very beneficial to know that since the Additive Identity Property states that any number added to 0 will always be that number, you will always know that any two bags of buy-one-get-one-free potatoes will always be $6.25.