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# Algebraic Manipulation

Gauss, Ramanujan, and a pantheon of other mathematicians have given us algebraic manipulation tools way beyond what's taught in school. Learn what they knew.

# Algebraic Manipulation Problem Solving

If $$a$$ and $$b$$ are real numbers such that $a^2+b^2=8, ab=3,$ what is the value of $\frac{a^7-ab^6-ba^6+b^7}{a+b}?$

Suppose that $$x$$ and $$y$$ are positive real numbers satisfying $$x^2+y^2 = 5xy$$. Then $$\frac{x-y}{x+y}$$ can be written as $$\sqrt{\frac{a}{b}}$$, where $$a$$ and $$b$$ are coprime positive integers. Find $$a+b$$.

$$a$$, $$b$$ and $$c$$ are real numbers such that $a+b+c \neq 0, a^3+b^3+c^3=12, abc=4.$ What is the value of $$(a+b)(b+c)(c+a)?$$

If $$A-B=-4$$ and $$AB=62$$, what is the value of $$A^2+B^2$$?

Given $$a+b+c=4$$, $$a^2+b^2+c^2=24$$ and $$a^3+b^3+c^3=6,$$ what is the value of $ab(a+b)+bc(b+c)+ca(c+a)?$

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