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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