Question

Identify and label each of the mathematical properties below.

(Commutative, Associative, Identity, or Distributive)
a + b = b + a
(a • b) • c = a • (b • c)
(a + b) + c = a + (b + c)
a • b = b • a
n + 0 = n
n × 1 = n
c(a + b) = ca + cb
n ÷ 1 = n
n − 0 = n

Simplify each of the expressions below using any combination of the properties above and combining like terms.
3x + 5 + 6x - 2(2 + 6x)
5(3x + 3)
3x + 5 - 6x + 16
x^2y + 2xy^2 + 3x^2y - 6xy^2 + 4

2(2 + 6x) - 5(3x + 3)
8(-14w + 10) - 19(10 + 4q) - 13q
4n - 3(-9n + 13) - 19 - 7(-17z + 13)

Answer 1

The mathematical properties mentioned in the question are the Commutative, Associative, Identity, and Distributive properties. These properties allow us to simplify mathematical expressions. To simplify the given expressions, we can apply these properties and combine like terms.

Step-by-step explanation:

The mathematical properties are:

• a + b = b + a - Commutative property
• (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c) - Associative property
• (a + b) + c = a + (b + c) - Associative property
• a ⋅ b = b ⋅ a - Commutative property
• n + 0 = n - Identity property
• n ⋅ 1 = n - Identity property
• c(a + b) = ca + cb - Distributive property
• n ÷ 1 = n - Identity property
• n - 0 = n - Identity property

Now, let's simplify the given expressions using these mathematical properties:

1) 3x + 5 + 6x - 2(2 + 6x)

1. Apply the distributive property: -2(2 + 6x) = -4 - 12x
2. Combine like terms: 3x + 6x + 5 - 4 - 12x = 9x + 1

2) 5(3x + 3)

1. Apply the distributive property: 5(3x) + 5(3) = 15x + 15

3) 3x + 5 - 6x + 16

1. Combine like terms: 3x - 6x + 5 + 16 = -3x + 21

4) x^2y + 2xy^2 + 3x^2y - 6xy^2 + 4

1. Combine like terms: x^2y + 3x^2y - 6xy^2 + 2xy^2 + 4 = 4x^2y - 4xy^2 + 4

5) 2(2 + 6x) - 5(3x + 3)

1. Apply the distributive property: 2(2) + 2(6x) - 5(3x) - 5(3) = 4 + 12x - 15x - 15
2. Combine like terms: 12x - 15x + 4 - 15 = -3x - 11

6) 8(-14w + 10) - 19(10 + 4q) - 13q

1. Apply the distributive property: 8(-14w) + 8(10) - 19(10) - 19(4q) - 13q = -112w + 80 - 190 - 76q - 13q
2. Combine like terms: -112w - 89 - 89q

7) 4n - 3(-9n + 13) - 19 - 7(-17z + 13)

1. Apply the distributive property: 3(-9n + 13) = -27n + 39 and 7(-17z + 13) = -119z + 91
2. Combine like terms: 4n + 27n - 39 - 19 + 119z - 91