Distributive Property

Number properties are fundamental rules and characteristics that govern how numbers behave under various mathematical operations. Understanding these properties is crucial for simplifying expressions, solving equations, and building a strong foundation in mathematics.

Summary of Number Properties

The following table summarizes the number properties for addition and multiplication: Commutative, Associative, Distributive, Identity and Inverse. Scroll down the page for examples, explanations and solutions.

Worksheets
Practice your skills with the following worksheets:
Printable & Online Numbers Worksheets
Printable & Online Algebra Worksheets

Basic Properties of Addition and Multiplication:
Commutative Property: The order in which numbers are added or multiplied does not change the result.
Addition: a+b=b+a (e.g., 3+5=5+3)
Multiplication: a×b=b×a (e.g., 2×7=7×2)
Check out this lesson on Commutative Property

Associative Property: The way numbers are grouped when adding or multiplying does not change the result.
Addition: (a+b)+c=a+(b+c) (e.g., (1+2)+3=1+(2+3))
Multiplication: (a×b)×c=a×(b×c) (e.g., (4×5)×6=4×(5×6))
Check out this lesson on Associative Property

Identity Property:
Addition: There exists an additive identity, 0, such that a+0=a (e.g., 9+0=9).
Multiplication: There exists a multiplicative identity, 1, such that a×1=a (e.g., 6×1=6).
Check out this lesson on Identity Property

Inverse Property:
Addition: For every number a, there exists an additive inverse (opposite), −a, such that a+(−a)=0 (e.g., 8+(−8)=0).
Multiplication: For every non-zero number a, there exists a multiplicative inverse (reciprocal),
\( \frac{1}{a} \), such that \( a \times \frac{1}{a} \) = 1 (e.g., 3 × \( \frac{1}{3} \) = 1

Distributive Property: Multiplication distributes over addition and subtraction. Distributive property allows you to remove the parenthesis (or brackets) in an expression. Multiply the value outside the brackets with each of the terms in the brackets.
a×(b+c)=(a×b)+(a×c) (e.g., 2×(3+4)=(2×3)+(2×4))
a×(b−c)=(a×b)−(a×c) (e.g., 5×(7−2)=(5×7)−(5×2))

For example:	4(a + b) = 4a + 4b
	7(2c – 3d + 5) = 14c – 21d + 35

What happens if you need to multiply (a – 3)(b + 4)?
You do the same thing but with one value at a time.

For example:
Multiply a with each term to get a × b + 4 × a = ab + 4a

Then, multiply 3 with each term to get “ –3b – 12” (take note of the sign operations).
Put the two results together to get “ab + 4a – 3b – 12”
Therefore, (a – 3)(b + 4) = ab + 4a – 3b – 12

The following video shows more examples of the distributive property.

• Show Step-by-step Solutions

The following video will discuss the number properties for Algebra.

• Show Step-by-step Solutions

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