Cambridge Maths Academy

Pure mathematics Year 1 Here's a short introduction to Pure maths 1 Chapter 1 Algebraic expressions. 1.1 Index laws: Multiply and divide integer powers. 1.2 Expanding brackets: Expand a single term over brackets and collect like terms. Expand the product of two or three expressions. 1.3 Factorising: Factorise linear, quadratic and simple cubic expressions. 1.4 Negative and fractional indices: Kn..

We have: $$ \begin{align} 1^1&=1 &&& 0^1&=0 \\ 1^0&=1 &&& 0^0&=1 \end{align} $$ Are you surprised by $0^0=1$? Proof. To prove this, we consider $$ \begin{align} y=x^x \end{align} $$ and take the limit $x\rightarrow0$. It is not straightforward to do this directly with $x^x$ so we take the (natural or any) logarithm on both sides: $$ \begin{align} \ln y=\ln x^x=x\ln x \end{align} $$ Let $x=e^{-n}..