A Level Pure Maths
Cheat Sheet

The following is a list of Rules, Results & Formulae that are not given in the exam booklet. Refer to it in order to help you memorise such rules and equations!

Basics

The Quadratic Formula:

The roots of the quadratic equation: ax^2+bx+c=0, where a, b &  c are constants, and a\neq 0 are given by:

\boxed{x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}}

Surds

Rule 1:

\boxed{\sqrt{ab}=\sqrt{a}\sqrt{b}}

Rule 2:

\boxed{\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}}

Rule 3 (follows from Rule 1):

\boxed{\sqrt{a}\sqrt{a}=a}

Indices

Rule 1:

\boxed{a^m a^p=a^{m+p}}

Rule 2:

\boxed{\frac{a^m}{a^p}=a^{m-p}}

Rule 3:

\boxed{(a^m)^p=a^{mp}}

Result 1:

\boxed{a^0=1}

Result 2:

\boxed{a^{-p}=\frac{1}{a^p}}\text{}

Rule 4:

\boxed{a^{\frac{1}{p}}=\sqrt[p]{a}}

Rule 5:

\boxed{a^{\frac{m}{p}}=\sqrt[p]{a^m}}\text{ or }\boxed{a^{\frac{m}{p}}={(\sqrt[p]{a})}^m}

Logarithms

Relation between logs and indices:

\boxed{a^x=b\iff \log_a(b)=x} where b is a positive number.

The Common Log:

\boxed{\log_{10}=\log}

The Natural Log:

\boxed{\log_e=\ln}

Result 1:

\boxed{\log_a a=1} for any positive number a

Result 2:

\boxed{\log_a 1=0} for any positive number a

Rule 1:

\boxed{\log_c{ab}=\log_c{a}+\log_c{b}} where a, b & c are positive numbers

Rule 2:

\boxed{\log_c{\frac{a}{b}}=\log_c{a}-\log_c{b}} where a, b & c are positive numbers

Rule 3:

\boxed{\log_c{a^b}=b\log_c{a}} where a, b & c are positive numbers