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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.

Problem Solving - Basic

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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