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Algebraic Manipulation

Algebraic Manipulation Problem Solving


If aa and bb are real numbers such that a2+b2=8,ab=3,a^2+b^2=8, ab=3, what is the value of a7ab6ba6+b7a+b?\frac{a^7-ab^6-ba^6+b^7}{a+b}?

Suppose that xx and yy are positive real numbers satisfying x2+y2=5xyx^2+y^2 = 5xy. Then xyx+y \frac{x-y}{x+y} can be written as ab \sqrt{\frac{a}{b}} , where aa and bb are coprime positive integers. Find a+ba+b.

aa, bb and cc are real numbers such that a+b+c0,a3+b3+c3=12,abc=4.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)?(a+b)(b+c)(c+a)?

If AB=4A-B=-4 and AB=62AB=62, what is the value of A2+B2A^2+B^2?

Given a+b+c=4a+b+c=4, a2+b2+c2=24a^2+b^2+c^2=24 and a3+b3+c3=6,a^3+b^3+c^3=6, what is the value of ab(a+b)+bc(b+c)+ca(c+a)?ab(a+b)+bc(b+c)+ca(c+a)?


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