Help with GCSE/A-Level learning materials

Question

I'm trying to help my nephew with his Maths for GCSE and A-Level (UK). It's been many years since I've done this. All the books I came across provide the formulae but don't explain how the authors arrived at those formulae. I'm searching for a suitable book(s) or any other sources that would provide the formulae but, most importantly, explain the logic used to derive them. I illustrate below the type of explanations I am after using a few examples from laws of indices. Sorry for the layout, TeX beginner here.

Multiplying powers

$$4^3*4^5=$$ $$(4*4*4)*(4*4*4*4*4)=$$ $$4*4*4*4*4*4*4*4=$$ $$4^8=$$ $$4^{3+5}$$

Dividing powers $$5^{7}\div5^{3}=$$ $$\frac{5*5*5*5*5*5*5}{5*5*5}=$$ $$\frac{5*5*5*5}{1}=$$ $$\frac{5^{4}}{1}=$$ $$\frac{5^{7-3}}{1}=$$ $$5^4$$

Raising a power to another power: $$(3^{2})^{4}=3^2*3^2*3^2*3^2=(3*3)*(3*3)*(3*3)*(3*3)=3^8=3^{2*4}$$

Raising to the power of $0$ $$a^0\iff a^{b-b}\iff \frac{a^b}{a^b}=1\therefore a^0=1$$

Negative power denotes reciprocal

$$4^{-2}\iff 4^{0-2}\iff \frac{4^0}{4^2}=\frac{1}{4^2}$$

Fractional power denotes square roots and cube roots with the important generalisation of power m/n. We notice the better shorthand offered by the surd notation.

$$9^{\frac{1}{2}} \iff \sqrt{9}=3$$ $$8^{\frac{1}{3}} \iff \sqrt[3]{8}=2$$ $$a^{\frac{m}{n}} \iff a^{m*\frac{1}{n}} \iff (a^{m})^\frac{1}{n}\iff \sqrt[n]{m}$$ $$a^{\frac{m}{n}} \iff a^{m*\frac{1}{n}} \iff (a^{\frac{1}{n}})^m\iff (\sqrt[n]{a})^m$$

Answer 1

As a maths tutor in the UK, I use: Essential Prealgebra by Chris McMullen, for integer powers. However, the book only contains a small section on fractional indices with denominator $2,$ i.e. square roots. It doesn't cover cubed roots and so on. The explanations at the start of each chapter helps many of my students improve, although some of them don't like this book or find it too challenging.

Additionally or alternatively, a book that thoroughly covers all topics for GCSE with exercise on every topic, including fractional powers, is: Maths for GCSE and IGCSE® Textbook - Higher by CGP. This book is great: it is useful for anyone taking higher level GCSE or iGCSE maths, as it covers the entire GCSE syllabus. However, I find that many students are intimidated by the size of this 500+ page book, and so you can also look into Collins GCSE maths books like this one, although I find these to not be thorough enough nor have enough questions for students to retain long-term memory of what they need to know.

An easier solution for the fractional indices might be to search "fractional indices GCSE" on youtube and get them to watch the first few videos that comes up, and see if any of them help. For example, I think this video is good as it encourages active participation rather than passive learning. Then get the student to have a go at this worksheet, or do the exercises in one of the GCSE books above. You can find that worksheet and the solutions on https://www.mathsgenie.co.uk/ .