Simplification
🔒 Log in to trackmedium importance~2 Q in Tier 126 formulas⚡ 14 shortcuts5 subtopics
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BODMAS, 'of', brackets & vinculum
'of' before division
a \div b \text{ of } c = \frac{a}{b \times c}
÷ and × left to right
a \div b \times c = \frac{a}{b} \times c
Removing a bracket after minus
a - (b - c) = a - b + c
Continued fraction (bottom-up)
a + \cfrac{1}{b + \cfrac{1}{c}} = a + \frac{c}{bc + 1}
Algebraic identities in numerical simplification
Square of sum / difference
(a \pm b)^2 = a^2 \pm 2ab + b^2
Difference of squares
a^2 - b^2 = (a+b)(a-b)
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)
Cube of sum
(a+b)^3 = a^3 + b^3 + 3ab(a+b)
Three cubes
a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca)
Zero-sum case
a + b + c = 0 \Rightarrow a^3 + b^3 + c^3 = 3abc
Surds & indices
Product / quotient
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 & 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,\ x > y
Reciprocal of a unit surd
x = a + \sqrt{b},\ a^2 - b = 1 \Rightarrow \frac{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 + \sqrt{n + \cdots}}} = \frac{1 + \sqrt{1 + 4n}}{2}
Infinite radical (minus)
\sqrt{n - \sqrt{n - \sqrt{n - \cdots}}} = \frac{-1 + \sqrt{1 + 4n}}{2}
Infinite nested product
\sqrt{x\sqrt{x\sqrt{x \cdots}}} = x
Finite nested product
\sqrt{x\sqrt{x\sqrt{x}}} = x^{7/8}
n roots: exponent (2ⁿ − 1)/2ⁿ
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},\ 16.67\% = \tfrac{1}{6},\ 33.33\% = \tfrac{1}{3},\ 37.5\% = \tfrac{3}{8}