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Distributing Rules

Distributing Rules

Algebra rules for distributing expressions. See factoring rules as well.

 

A. Multiplication

1. addition:  a(b + c) = ab + ac  and  (b + c)a = ba + ca

2. subtraction:  a(b – c) = ab – ac  and  (b – c)a = ba – ca

3. FOIL:  (a + b)(c + d) = ac + ad + bc + bd

Careful!!

a(bc) ≠ ab·ac

 

B. Division

1. addition:  Math formula showing division distributes over addition: (a + b) / c = a/c + b/c

2. subtraction:  Math formula showing (a minus b) divided by c equals a/c minus b/c

Careful!!

Math warning: ab/c ≠ (a/c)·(b/c)

 

C. Exponents   (a ≥ 0, b ≥ 0)

1. multiplication:  (ab)x = axbx

2. division:  Math formula showing exponent distribution over division: (a/b)^x = a^x / b^x     (b ≠ 0)

Careful!!

(a + b)x ≠ ax + bx

(a – b)x ≠ ax – bx

 

D. Roots   (x ≥ 0, y ≥ 0)

1. multiplication:  The nth root of (xy) equals the nth root of x times the nth root of y: ⁿ√(xy) = ⁿ√x · ⁿ√y

2. division:  nth root of (x/y) = (nth root of x) / (nth root of y)     ( y ≠ 0)

Careful!!

The nth root of (x + y) ≠ nth root of x + nth root of y

The nth root of (x minus y) is not equal to the nth root of x minus the nth root of y

 

E. Logarithms   (x > 0, y > 0, a > 0, a ≠ 1)

1. multiplication:  loga (xy) = loga x + loga y

2. division:  log base a of (x/y) = log base a of x minus log base a of y

3. powers:  loga (xp) = p loga x

Careful!!

loga (x + y) ≠ loga x + loga y

loga (x – y) ≠ loga x – loga y

 

F. Trig

1. sin(x ± y) = sin x cos y ± cos x sin y

2. cos(x ± y) = cos x cos y ∓ sin x sin y

3. tan(x ± y) = (tan x ± tan y) / (1 ∓ tan x tan y), the tangent addition and subtraction formula

 

See also

Binomial theorem, fraction rules, exponent rules, radical rules, square root rules, logarithm rules, sum/difference identities

Key Formula

a(b+c)=ab+aca(b + c) = ab + ac
Where:
  • aa = The factor being distributed (multiplied) to each term inside the parentheses
  • bb = The first term inside the parentheses
  • cc = The second term inside the parentheses

Worked Example

Problem: Use the distributive property to expand and simplify 5(2x + 3).
Step 1: Identify the factor outside the parentheses and the terms inside.
a=5,b=2x,c=3a = 5,\quad b = 2x,\quad c = 3
Step 2: Multiply the outside factor by the first term inside.
5⋅2x=10x5 \cdot 2x = 10x
Step 3: Multiply the outside factor by the second term inside.
5⋅3=155 \cdot 3 = 15
Step 4: Combine the two products.
5(2x+3)=10x+155(2x + 3) = 10x + 15
Answer: 5(2x + 3) = 10x + 15

Another Example

This example shows distributing an exponent over multiplication, illustrating that distribution rules extend beyond basic multiplication. It also highlights the common error of distributing an exponent over addition.

Problem: Distribute the exponent in (3y)⁴ and simplify.
Step 1: Recognize this uses the exponent distribution rule: (ab)ˣ = aˣ · bˣ.
(3y)4=34⋅y4(3y)^4 = 3^4 \cdot y^4
Step 2: Evaluate the numerical power.
34=813^4 = 81
Step 3: Write the final simplified form.
(3y)4=81y4(3y)^4 = 81y^4
Step 4: Warning: this rule works because 3y is a product. You CANNOT distribute an exponent over a sum. For instance, (3 + y)⁴ ≠ 3⁴ + y⁴.
(3+y)4≠81+y4(3 + y)^4 \neq 81 + y^4
Answer: (3y)⁴ = 81y⁴

Frequently Asked Questions

What is the difference between distributing and factoring?
Distributing and factoring are inverse operations. Distributing expands an expression by multiplying a factor across each term inside parentheses, as in a(b + c) = ab + ac. Factoring reverses this process by pulling a common factor out, turning ab + ac back into a(b + c). Knowing one direction helps you work in the other.
Can you distribute exponents over addition or subtraction?
No. A very common mistake is writing (a + b)ˣ = aˣ + bˣ, but this is incorrect. Exponents distribute only over multiplication and division: (ab)ˣ = aˣbˣ and (a/b)ˣ = aˣ/bˣ. To expand (a + b)ˣ, you need the Binomial Theorem or must multiply the expression out manually.
When do you use the distributive property?
You use the distributive property whenever you need to remove parentheses in an algebraic expression, simplify equations for solving, or multiply polynomials. It appears constantly in algebra, from combining like terms to applying FOIL for two binomials. It is also the foundation for distributing roots, logarithms, and exponents over products.

Distributing vs. Factoring

DistributingFactoring
DirectionExpands: breaks parentheses apartCompresses: groups terms into parentheses
Formulaa(b + c) → ab + acab + ac → a(b + c)
PurposeSimplify or remove grouping symbolsFind common factors or solve equations
When to useWhen you need to expand and combine like termsWhen you need to solve equations or simplify fractions

Why It Matters

Distributing rules appear in nearly every algebra course and carry forward into calculus, physics, and engineering. You use them every time you expand expressions, solve equations, or simplify logarithmic and exponential forms. Knowing which operations truly distribute—and which do not—prevents errors that can derail an entire solution.

Common Mistakes

Mistake: Distributing an exponent or root over addition/subtraction, e.g., writing (a + b)² = a² + b² or √(a + b) = √a + √b.
Correction: Exponents and roots distribute only over multiplication and division. For (a + b)², expand using FOIL or the identity a² + 2ab + b². For √(a + b), the expression generally cannot be simplified further.
Mistake: Distributing a logarithm over addition, e.g., writing log(x + y) = log x + log y.
Correction: The log rule log(xy) = log x + log y applies to products, not sums. There is no simple rule for log(x + y). The expression log x + log y equals log(xy), not log(x + y).

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