Algebra Formulas
The following table contains alternative ways of expressing algebraic functions.
General
a^{x} = 1 / a^{x} 

a^{1/x} = ^{x}√a 

a^{1/x} = 1 / ^{x}√a 

(a.b)^{x} = a^{x} . b^{x} 

(a^{x})^{y} = a^{x.y} 

aˣ⁺ʸ = aˣ . aʸ 
e.g.: a²˙⁵ = a² . a⁰˙⁵ 
xⁿ/yⁿ + 1 = (xⁿ + yⁿ) / yⁿ 

(x  a) / A = x/A  a/A 

(x²  a²) = (x  a).(x + a) 

Simplify: (x + b)/(x²  a²) (x + b)/(x²  a²) = (x + b) / [(x  a).(x + a)] Continue in order to prove the above: 

Factorial: e.g. 5! = 5x4x3x2x1 
Binomial
(a+b)^{n} = a^{n} + ^{n}C_{1}.a^{(n1)}.b + ^{n}C_{2}.a^{(n2)}.b2 + ..... + ^{n}C_{r}.a^{(nr)}.b^{r} + ..... b^{n} 
If; 0 = ax² + bx + c 
Simple and Compound Interest
The following table contains formulas for calculating simple and compound interest.
Where: P = the principal sum, p = percentage interest, n = payment term (years), q = payments per year, I = interest paid over full term, m = amount of each payment & P_{n} = total amount paid over 'n' years
Simple
I = P.p.n 
P_{n} = I + P 
Compound
I = P_{n}  P 
P_{n} = P.(1 + p/q)^{n.q} 
m = P_{n} / n.q 
NetPresentValue
V = P_{n} / (1+p/q)^{n.q} 
Discount(simple) = P_{n}(simple)  V 
Discount(compound) = P_{n}(compound)  V 
Progressions
The following table contains the formulas for arithmetic and geometric progressions.
Note: r = 2^{nd} term ÷ 1^{st} term, d = 2^{nd} term  1^{st} term, n = number of terms
Arithmetic
n^{th} term = 1^{st} term + d.(n1) 
Σn terms = n.[2 . 1^{st} term.(n1)] / 2 
Geometric
n^{th} term = 1^{st} term . r^{(n1)}) 
Σn terms = 1^{st} term . (1  r^{n}) / (1  r) [r < 1] 
Σn terms = 1^{st} term . (r^{n}  1) / (r  1) [r > 1] 
Logarithms
The following table contains alternative ways of expressing logarithmic functions.
Note: 'base' refers to the logarithmic base, which can be any positive number
The most common bases are 10 and 2.71828182845905 (the base for natural logs, normally written thus 'ln(x)')
e = 1 + 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + ..... = 2.71828182845905 
e^{x} = 1 + x¹/1 + x²/2 + x³/3 + x⁴/4 + x⁵/5 + x⁶/6 + x⁷/7 + .... 
e^{x} = 1  x¹/1 + x²/2  x³/3 + x⁴/4  x⁵/5 + x⁶/6  x⁷/7 + .... 
y = Log_{base}(base^{y}) 
Log x = Log_{base}(x) 
log_{base}(x) = log_{a}(x) / log_{a}(base) 
base^{(a+b+c)} = base^{(a)}.base^{(b)}.base^{(c)} 
Log_{base}(x)  Log_{base}(y) = log_{10}(base).x/y 
Log_{base}(x^{a}) = a.Log_{base}(x) 
Log_{base}(x)  Log_{base}(y) = z 
1/exp(x) = exp(x) 
e^{ln(x)} = x 
Xn = Y 
Determinants
Determinants are a means of solving simultaneous equations, e.g.
a₁.x + b₁.y + c₁.z = 0
a₂.x + b₂.y + c₂.z = 0
a₃.x + b₃.y + c₃.z = 0
which can be written thus:
ǀa₁,b₁,c₁ǀ
ǀa₂,b₂,c₂ǀ
ǀa₃,b₃,c₃ǀ
The following table contains the procedure for solving determinants.
2ᴺᴰ Order a₁.w + b₁.x = 0 a₁,b₁ = a₁.b₂  a₂.b₁ To solve a 2ᴺᴰ Order equation you perform the calculation as shown above 
3ᴿᴰ Order a₁.w + b₁.x + c₁.y = 0 a₁,b₁,c₁ = a₁.b₂.c₂  a₂.b₁.c₁ + a₃.b₁.c₁ To solve a 3ᴿᴰ Order equation you convert to 2ᴺᴰ Order equations as shown above then solve 2ᴺᴰ Order equations 
4ᵀᴴ Order a₁.w + b₁.x + c₁.y + d₁.z = 0 a₁,b₁,c₁.d₁ = a₁.b₂.c₂.d₂  a₂.b₁.c₁.d₁ + a₃.b₁.c₁.d₁  a₄.b₁.c₁.d₁ To solve a 4ᵀᴴ Order equation you convert to 3ᴿᴰ Order equations as shown above, then convert to 2ᴺᴰ Order equations then solve 2ᴺᴰ Order equations 
The same procedure may be followed for all subsequent Order equations: 5ᵀᴴ, 6ᵀᴴ, 7ᵀᴴ, etc.
Fig 1. Finding x and y
Slope
A calculation method to find a point on a graph of positive or negative slope
Positive Slope
with reference to Fig 1
x = (x₂x₁).(yy₁)/(y₂y₁) + x₁
y = (y₂y₁).(xx₁)/(x₂x₁) + y₁
Negative Slope
Fig 2. Finding x and y
with reference to Fig 2
x = (x₂x₁).(yy₂)/(y₁y₂) + x₁
y = (y₁y₂).(xx₁)/(x₂x₁) + y₂
Slope
A calculation method to find a point on a graph of positive or negative slope
Geometry
Properties of a Sphere
Circumference: C = 2.π.r
Surface Area: A = 4.π.r²
Volume: V = ⁴/₃.π.r³
Distance between 'n' equally spaced points on the surface:
Arc: d = π.A / C.n = 6.π.V / A.n
Linear: ℓ = r.Sin(½.d/r)
Further Reading
You will find further reading on this subject in reference publications^{(19)}