Patterns in Addition and Subtraction

Study Notes

Commutative Property: The order of addends or subtrahends does not change the result.
- Example: 2 + 3 = 3 + 2
- Example: 5 - 2 = 2 - 5 (not true, but can be rewritten as 5 - 2 = -(2 - 5))
Associative Property: The order in which numbers are added or subtracted does not change the result.
- Example: (2 + 3) + 4 = 2 + (3 + 4)
- Example: (5 - 2) - 3 = 5 - (2 + 3)
Distributive Property: Multiplication can be distributed over addition and subtraction.
- Example: 2 × (3 + 4) = 2 × 3 + 2 × 4
- Example: 2 × (5 - 2) = 2 × 5 - 2 × 2

Commutative Property: The order of factors does not change the product.
- Example: 2 × 3 = 3 × 2
Associative Property: The order in which numbers are multiplied does not change the product.
- Example: (2 × 3) × 4 = 2 × (3 × 4)
Distributive Property: Multiplication can be distributed over addition and subtraction.
- Example: 2 × (3 + 4) = 2 × 3 + 2 × 4
- Example: 2 × (5 - 2) = 2 × 5 - 2 × 2
Multiplication by Zero: Any number multiplied by 0 is 0.
- Example: 2 × 0 = 0
Division by Zero: Undefined, as it is not possible to divide a number by 0.

Exponential Rules:
- Product of Powers: a^m × a^n = a^(m+n)
- Power of a Product: (ab)^m = a^m b^m
- Power of a Power: (a^m)^n = a^(mn)
Properties of Roots:
- Square Root: √(ab) = √a × √b
- Cube Root: ³√(ab) = ³√a × ³√b

These patterns and properties help simplify and solve numerical operations, making it easier to work with numbers.

The order of addends or subtrahends does not change the result, known as the Commutative Property.
The order in which numbers are added or subtracted does not change the result, known as the Associative Property.
Multiplication can be distributed over addition and subtraction, known as the Distributive Property.

The order of factors does not change the product, known as the Commutative Property.
The order in which numbers are multiplied does not change the product, known as the Associative Property.
Multiplication can be distributed over addition and subtraction, known as the Distributive Property.
Any number multiplied by 0 is 0, known as Multiplication by Zero.
Division by 0 is undefined.

The product of powers is a^(m+n) when a^m is multiplied by a^n, known as the Product of Powers.
The power of a product is a^m b^m when (ab) is raised to the power of m, known as the Power of a Product.
The power of a power is a^(mn) when a^m is raised to the power of n, known as the Power of a Power.

The square root of a product is √a × √b when √(ab) is calculated, known as the Square Root.
The cube root of a product is ³√a × ³√b when ³√(ab) is calculated.