Math · Multiplication

Properties of Multiplication
Commutative, Associative & Distributive

Learn the 5 properties of multiplication with examples: commutative, associative, identity, zero and distributive. For elementary and middle school with exercises.

⚡ RESPUESTA RÁPIDA

The 5 properties of multiplication: Commutative: a×b=b×a. Associative: (a×b)×c=a×(b×c). Identity: a×1=a. Zero: a×0=0. Distributive: a×(b+c)=ab+ac.

The 5 Properties with Examples

Commutative: a × b = b × a

Order doesn't matter. 4×6 = 6×4 = 24. Cuts memorization in half.

Associative: (a×b)×c = a×(b×c)

Grouping doesn't matter. (2×5)×6 = 2×(5×6) = 60. Group to make 10s.

Identity: a × 1 = a

Multiplying by 1 doesn't change the number. 7×1=7. One is the multiplicative identity.

Zero Property: a × 0 = 0

Any number times zero equals zero. 999×0=0.

Distributive: a×(b+c) = ab+ac

3×(4+5) = 3×4+3×5 = 27. Key for mental math and algebra.

Identify the Property — 10 Exercises

6×7=7×6
Commutative
(2×5)×4=2×(5×4)
Associative
8×1=8
Identity
15×0=0
Zero
3×(6+4)=18+12
Distributive
a×b=b×a
Commutative
n×1=n
Identity
(x×y)×z=x×(y×z)
Associative
0×999=0
Zero
5×(10+3)=50+15
Distributive

See also properties of addition for the parallel rules with addition.

Real-World Examples of Each Property

Commutative — Shopping

Buying 3 packs of 6 items = buying 6 packs of 3 items = 18 total. 3×6=6×3=18.

Distributive — Mental Math

15% of $80: 10% of $80 + 5% of $80 = $8 + $4 = $12. Faster than 0.15×80 directly.

Zero Property — Area

A rectangle with height 0 has area = 0, no matter how wide it is. L×0=0.

15 Practice Exercises

7×8=8×?
7 (comm.)
(3×4)×5=3×(4×?)
5 (assoc.)
9×1
9 (identity)
0×47
0 (zero)
6×(10+3)=?+18
60 (dist.)
5×(20+4)
120
(2×6)×5=2×(6×5)=?
60
12×1
12
0×1000
0
4×(25+5)
120
a×b=b×?
a
n×0
0
(x×y)×1=?
x×y
3×(100+7)
321
a×1
a

Preguntas Frecuentes

Does the distributive property work with subtraction?

Yes. a×(b−c)=ab−ac. Example: 5×(10−3)=50−15=35. Same as 5×7=35.

Why is the zero property important in algebra?

It helps solve equations. If (x−3)(x+2)=0, then either x−3=0 or x+2=0, giving x=3 or x=−2.

Also useful