Simplification
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BODMAS, 'of', brackets & vinculum
'Of' before division
a \div b \text{ of } c = a \div (b \times c)
Division and multiplication left to right
a \div b \times c = \dfrac{a}{b} \times c
Minus before a bracket
a - (b - c) = a - b + c
Continued fraction
a + \dfrac{1}{b + \dfrac{1}{c}} = a + \dfrac{c}{bc + 1}
Algebraic identities in numerical simplification
Sum of cubes
a^3 + b^3 = (a+b)(a^2 - ab + b^2)
Difference of cubes
a^3 - b^3 = (a-b)(a^2 + ab + b^2)
Difference of squares
a^2 - b^2 = (a+b)(a-b)
Cube of sum
(a+b)^3 = a^3 + b^3 + 3ab(a+b)
Squares combine
(a+b)^2 + (a-b)^2 = 2(a^2+b^2)
Squares subtract
(a+b)^2 - (a-b)^2 = 4ab
Zero-sum cubes
a+b+c = 0 \Rightarrow a^3+b^3+c^3 = 3abc
Square-sum from pair facts
a^2 + b^2 = (a+b)^2 - 2ab
Surds & indices
Same base
a^m \cdot a^n = a^{m+n},\quad \frac{a^m}{a^n} = a^{m-n}
Power of power
(a^m)^n = a^{mn},\quad (ab)^n = a^n b^n
Zero and negative index
a^0 = 1,\quad a^{-n} = \frac{1}{a^n}
Fractional index
a^{p/q} = \sqrt[q]{a^p}
Rationalisation
\frac{1}{\sqrt{a} \pm \sqrt{b}} = \frac{\sqrt{a} \mp \sqrt{b}}{a - b}
Root of a surd
\sqrt{a \pm 2\sqrt{b}} = \sqrt{x} \pm \sqrt{y},\ x + y = a,\ xy = b
Reciprocal of a unit surd
x = a + \sqrt{b},\ a^2 - b = 1 \Rightarrow \tfrac{1}{x} = a - \sqrt{b}
Square roots & cube roots
Product rule
\sqrt{ab} = \sqrt{a}\,\sqrt{b},\quad \sqrt[3]{ab} = \sqrt[3]{a}\,\sqrt[3]{b}
Infinite radical, plus
\sqrt{n + \sqrt{n + \cdots}} = \dfrac{1 + \sqrt{1 + 4n}}{2}
Infinite radical, minus
\sqrt{n - \sqrt{n - \cdots}} = \dfrac{-1 + \sqrt{1 + 4n}}{2}
Infinite nested product
\sqrt{x\sqrt{x\sqrt{x \cdots}}} = x
Least add or subtract
\text{subtract } N - k^2,\quad \text{add } (k+1)^2 - N
Approximation
Percentage swap
x\% \text{ of } y = y\% \text{ of } x
Near-square root
\sqrt{n^2 + k} \approx n + \frac{k}{2n}
Percent-fraction anchors
12.5\% = \tfrac{1}{8},\quad 16.7\% = \tfrac{1}{6},\quad 33.3\% = \tfrac{1}{3},\quad 37.5\% = \tfrac{3}{8}