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Mathematics

Simplify :

(xax−b)a2−ab+b2×(xbx−c)b2−bc+c2×(xcx−a)c2−ca+a2\Big(\dfrac{x^a}{x^{-b}}\Big)^{a^2 - ab + b^2} \times \Big(\dfrac{x^b}{x^{-c}}\Big)^{b^2 - bc + c^2} \times \Big(\dfrac{x^c}{x^{-a}}\Big)^{c^2 - ca + a^2}

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Answer

Simplifying the expression :

⇒(xax−b)a2−ab+b2×(xbx−c)b2−bc+c2×(xcx−a)c2−ca+a2=(xa−(−b))a2−ab+b2×(xb−(−c))b2−bc+c2×(xc−(−a))c2−ca+a2=x(a+b)(a2−ab+b2)×x(b+c)(b2−bc+c2)×x(c+a)(c2−ca+a2)=xa3+b3×xb3+c3×xc3+a3=xa3+b3+b3+c3+c3+a3=x2a3+2b3+2c3=x2(a3+b3+c3).\Rightarrow \Big(\dfrac{x^a}{x^{-b}}\Big)^{a^2 - ab + b^2} \times \Big(\dfrac{x^b}{x^{-c}}\Big)^{b^2 - bc + c^2} \times \Big(\dfrac{x^c}{x^{-a}}\Big)^{c^2 - ca + a^2} \\[1em] = (x^{a - (-b)})^{a^2 - ab + b^2} \times (x^{b - (-c)})^{b^2 - bc + c^2} \times (x^{c - (-a)})^{c^2 - ca + a^2} \\[1em] = x^{(a + b)(a^2 - ab + b^2)} \times x^{(b + c)(b^2 - bc + c^2)} \times x^{(c + a)(c^2 - ca + a^2)} \\[1em] = x^{a^3 + b^3} \times x^{b^3 + c^3} \times x^{c^3 + a^3} \\[1em] = x^{a^3 + b^3 + b^3 + c^3 + c^3 + a^3} \\[1em] = x^{2a^3 + 2b^3 + 2c^3} \\[1em] = x^{2(a^3 + b^3 + c^3)}.

Hence, (xax−b)a2−ab+b2×(xbx−c)b2−bc+c2×(xcx−a)c2−ca+a2=x2(a3+b3+c3).\Big(\dfrac{x^a}{x^{-b}}\Big)^{a^2 - ab + b^2} \times \Big(\dfrac{x^b}{x^{-c}}\Big)^{b^2 - bc + c^2} \times \Big(\dfrac{x^c}{x^{-a}}\Big)^{c^2 - ca + a^2} = x^{2(a^3 + b^3 + c^3)}.

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