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Mathematics

Simplify the following:

(xaxb)a2+ab+b2.(xbxc)b2+bc+c2.(xcxa)c2+ac+a2\Big(\dfrac{x^a}{x^b}\Big)^{a^2 + ab + b^2}.\Big(\dfrac{x^b}{x^c}\Big)^{b^2 + bc + c^2}.\Big(\dfrac{x^c}{x^a}\Big)^{c^2 + ac + a^2}

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Answer

Given,

⇒(xaxb)a2+ab+b2.(xbxc)b2+bc+c2.(xcxa)c2+ac+a2⇒(x(a−b))(a2+ab+b2).(x(b−c))(b2+bc+c2).(x(c−a))(c2+ca+a2)\Rightarrow \Big(\dfrac{x^a}{x^b}\Big)^{a^2 + ab + b^2}.\Big(\dfrac{x^b}{x^c}\Big)^{b^2 + bc + c^2}.\Big(\dfrac{x^c}{x^a}\Big)^{c^2 + ac + a^2} \\[1em] \Rightarrow (x^{(a - b)})^{(a^2 + ab + b^2)}.(x^{(b - c)})^{(b^2 + bc + c^2)}.(x^{(c - a)})^{(c^2 + ca + a^2)}

By formula,

(x3 - y3) = (x - y)(x2 + y2 + xy) we get,

⇒(x)a3−b3.(x)b3−c3.(x)c3−a3⇒xa3−b3+b3−c3+c3−a3⇒x0=1.\Rightarrow (x)^{a^3 - b^3}.(x)^{b^3 - c^3}.(x)^{c^3 - a^3} \\[1em] \Rightarrow x^{a^3 - b^3 + b^3 - c^3 + c^3 - a^3} \\[1em] \Rightarrow x^{0} = 1.

Hence, (xaxb)a2+ab+b2.(xbxc)b2+bc+c2.(xcxa)c2+ac+a2\Big(\dfrac{x^a}{x^b}\Big)^{a^2 + ab + b^2}.\Big(\dfrac{x^b}{x^c}\Big)^{b^2 + bc + c^2}.\Big(\dfrac{x^c}{x^a}\Big)^{c^2 + ac + a^2} = 1.

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