The following topics are considered to be necessary before attempting to learn this content for this module of Cambridge AS Pure Mathematics.
Transposition
This is a fancy term which means changing the subject of the formula.
Laws of indices
Any number raised to the power of $1$ is itself: $$ \begin{equation}\begin{aligned} a^1&=a\\ \end{aligned}\end{equation} $$
Any number raised to the power of zero ($0$) is $1$: $$ \begin{equation}\begin{aligned} a^0&=1\\ \end{aligned}\end{equation} $$
When multiplying two terms with the same base, it is the same as writing the base raised to the sum of the powers:
$$ \begin{equation}\begin{aligned} a^m\times a^n=a^{m+n}\\ \end{aligned}\end{equation} $$When we divide two terms with the same base, we can subtract the powers:
$$ \begin{equation}\begin{aligned} \frac{a^m}{a^n}=a^{m-n}\\ \end{aligned}\end{equation} $$When given a term with a power raised to another power, we can rewrite the base raised to the power of the product of the two ($2$) original powers:
$$ \begin{equation}\begin{aligned} (a^m)^n=a^{m\times n}\\ \end{aligned}\end{equation} $$When we raise a number to a negative power, it is the same as having the reciprocal of the number raised to the positive version of that power: $$ \begin{equation}\begin{aligned} a^{-n}&=\frac{1}{a^n}\\ \text{e.g. }a^{-1}&=\frac{1}{a}\\ \text{and }a^{-2}&=\frac{1}{a^2}\\ \end{aligned}\end{equation} $$
The square root (or cube root, etc.) of a number is simply the number raised to the reciprocal of the degree ($2$, $3$, etc.) of the radical:
$$ \begin{equation}\begin{aligned} \sqrt[n]{a}&=a^{\frac{1}{n}}\\ \text{e.g. }\sqrt[2]{a}&=a^{\frac{1}{2}}\\ \text{and }\sqrt[3]{a}&=a^{\frac{1}{3}}\\ \end{aligned}\end{equation} $$The little number on the root sign of the square/cube/etc. root is called the index/degree of the radical.
More generally, if we have a root with degree $n$ of a number raised to a power $m$, that is the same as the number raised to the power of $\frac{m}{n}$:
$$ \begin{equation}\begin{aligned} \sqrt[n]{a^m}=a^{\frac{m}{n}}\\ \end{aligned}\end{equation} $$When we have a fraction raised to a power, the result is the same as raising the numerator and denominator of the fraction to that power:
$$ \begin{equation}\begin{aligned} (\frac{a}{b})^{n}=\frac{a^n}{b^n}\\ \end{aligned}\end{equation} $$The same counts for the product of two numbers raised to a power:
$$ \begin{equation}\begin{aligned} (ab)^n=a^nb^n\\ \end{aligned}\end{equation} $$