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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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