Learn the 5 properties of multiplication with examples: commutative, associative, identity, zero and distributive. For elementary and middle school with exercises.
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.
Order doesn't matter. 4×6 = 6×4 = 24. Cuts memorization in half.
Grouping doesn't matter. (2×5)×6 = 2×(5×6) = 60. Group to make 10s.
Multiplying by 1 doesn't change the number. 7×1=7. One is the multiplicative identity.
Any number times zero equals zero. 999×0=0.
3×(4+5) = 3×4+3×5 = 27. Key for mental math and algebra.
See also properties of addition for the parallel rules with addition.
Buying 3 packs of 6 items = buying 6 packs of 3 items = 18 total. 3×6=6×3=18.
15% of $80: 10% of $80 + 5% of $80 = $8 + $4 = $12. Faster than 0.15×80 directly.
A rectangle with height 0 has area = 0, no matter how wide it is. L×0=0.
Yes. a×(b−c)=ab−ac. Example: 5×(10−3)=50−15=35. Same as 5×7=35.
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.
Commutative: 3×4×x=12x. Order of factors doesn't matter.
Distributive reversed: 6(x+y). This is factoring.
Zero property: either x−3=0 or x+3=0. So x=3 or x=−3.
It follows from the distributive property and the zero property. If (−1)×(−1)=−1, then (−1)×(−1)+(−1)×1=(−1)×(−1+1)=0, which means (−1)×(−1)=1.
No. 12÷4=3 but 4÷12=1/3. Division and subtraction are NOT commutative.
Only partially. Associative and distributive yes. But commutative NO — in matrix multiplication, A×B ≠ B×A in general.
a×b = b×a. Order does not change the product. 4×7 = 7×4 = 28.
(a×b)×c = a×(b×c). Grouping does not change the product.
a×(b+c) = a×b + a×c. Most important for algebra. 3×(4+5) = 12+15 = 27
a×1 = a (identity) | a×0 = 0 (zero property)
The distributive property is the foundation of algebra. When you expand (x+2)(x+3) or factor x²+5x+6, you are using it. Master this property and polynomial operations become straightforward.
The distributive property connects multiplication and addition — it is the engine behind algebra. When you see 5×(x+3), distributing gives 5x+15. This is used in every polynomial expansion, product of binomials, and factoring problem.
La propiedad distributiva es el puente entre multiplicación y suma — es la herramienta central del álgebra. Cuando factorizas, usas la distributiva al revés: ves 3x+12 y reconoces que viene de 3(x+4).
Ejercita la distributiva tanto en sentido directo (expandir) como inverso (factorizar). El 80% del álgebra de secundaria y preparatoria depende de esta propiedad. Si la dominas, todo lo demás es más fácil.
Usa conmutativa: si conoces 7×8=56, también sabes 8×7=56. Ahorra memorizaciones.
Usa distributiva para cálculo mental: 7×48=7×(50−2)=350−14=336.
Factoriza para simplificar: 15×8÷6 = 15×(8÷6) NO — aplica asociativa correctamente: (15÷6)×8=20.
La propiedad distributiva es la más poderosa de todas — es la base de la multiplicación de polinomios en álgebra, del cálculo mental rápido y de la simplificación de expresiones. Si solo aprendes una propiedad, aprende esta.
Las propiedades de la multiplicación son universales en matemáticas: aplican a enteros, decimales, fracciones, números negativos y expresiones algebraicas. La propiedad distributiva en particular es el puente entre la multiplicación y la suma, y es la base de todo el álgebra avanzada.