I'm a university student who never really studied maths in high school (I did the basic courses, but because I'm dyslexic I was to embarrassed to try the harder courses) now I'm getting back into it, and, while I'm perfectly fine with understanding the advanced concepts and grasping the theoretical side of things, I have a lot of trouble with the arithmetic in the harder questions. A lot of this is due to my not really knowing a lot of the techniques used in manipulating terms in complex fractions and just generally not having a good set of problem solving tools to fall back on.
Can you recommend any books that deal with arithmetic techniques for more advanced problems?
just to give you an idea, here are some examples of question formats that I would like to practice:
number 1 $$ \frac{x^{2}}{y}+\frac{y^{2}}{x} $$
where $x = 2 + \sqrt{3}$ and $y = 2 - \sqrt{3}$
Number 2
factorise $54x^3 + 16$
number 3
given that $$a^{2}+b^{2} = 7ab$$
express $$\left(\frac{a+b}{3}\right)^{2}$$
in terms of a and b. Hence show that $$\log\left (\frac{a+b}{3}\right )=\frac{1}{2}\left(\log a+\log b \right)$$
if would be especially great if it had lots of practice questions.
Thanks.
I recommend the text Algebra by I. M. Gelfand and A. Shen. It was one of a series of texts written for bright high school students in the Soviet Union who wished to supplement the high school curriculum through correspondence courses with professors at the University of Moscow. Working through this clearly written text will enhance your understanding of algebra and develop your problem-solving skills. The text is available in English translation.