Prove that: ( x^a x^b )^ a^2+ab+b^2 ( x^2 x^c )^ b^2+bc+c^2 ( x^c x^a )^ c^2+ca+a^2 =1

To prove ( x^a x^b )^ a^2+ab+b^2 ( x^2 x^c )^ b^2+bc+c^2 ( x^c x^a )^ c^2+ca+a^2 =1 Left hand side (LHS) = Right hand side (RHS) Considering LHS,= x^ a^3 +a^2b+ab^2 x^ a^2 b +ab^2+b^3 x^ b^3 +b^2c+bc^2 x^ b^2 c+bc^2+c^3 x^ c^3 +c^2a+ca^2 x^ c^2 a+ca^2+a^3 =x^ a^3 +a^2b+ab^2-(b^3+a^2bab^2) ^ b^3 +b^2c+bc^2-(c^3+b^2c+bc^2) ^ c^3 +c^2a+ca^2-(a^3+c^2a+ca^2) =x^ a^3 -b^3 ^ b^3-c^3 ^ c^3-a^3 =x^ a^3 -b^3+b^3-c^3-a^3 = x^0 =1 Or, Therefore, LHS = RHS Hence proved

Prove that: ( x^a x^b )^ a^2+ab+b^2 ( x^2 x^c )^ b^2+bc+c^2 ( x^c…

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Question

Prove that: $\Big(\frac{\text{x}^\text{a}}{\text{x}^\text{b}}\Big)^{\text{a}^2+\text{ab}+\text{b}^2}\times\Big(\frac{\text{x}^2}{\text{x}^\text{c}}\Big)^{\text{b}^2+\text{bc}+\text{c}^2}\times\Big(\frac{\text{x}^\text{c}}{\text{x}^\text{a}}\Big)^{\text{c}^2+\text{ca}+\text{a}^2}=1$

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Answer

To prove$\Big(\frac{\text{x}^\text{a}}{\text{x}^\text{b}}\Big)^{\text{a}^2+\text{ab}+\text{b}^2}\times\Big(\frac{\text{x}^2}{\text{x}^\text{c}}\Big)^{\text{b}^2+\text{bc}+\text{c}^2}\times\Big(\frac{\text{x}^\text{c}}{\text{x}^\text{a}}\Big)^{\text{c}^2+\text{ca}+\text{a}^2}=1$
Left hand side (LHS) = Right hand side (RHS) Considering LHS,$=\frac{\text{x}^{\text{a}^3}+\text{a}^2\text{b}+\text{ab}^2}{\text{x}^{\text{a}^2}{\text{b}}+\text{ab}^2+\text{b}^3}\times\frac{\text{x}^{\text{b}^3}+\text{b}^2\text{c}+\text{bc}^2}{\text{x}^{\text{b}^2}\text{c}+\text{bc}^2+\text{c}^3}\times\frac{\text{x}^{\text{c}^3}+\text{c}^2\text{a}+\text{ca}^2}{\text{x}^{\text{c}^2}\text{a}+\text{ca}^2+\text{a}^3}$
$=\text{x}^{\text{a}^3}+\text{a}^2\text{b}+\text{ab}^2-(\text{b}^3+\text{a}^2\text{b}\text{ab}^2)\\\times\text{x}^{\text{b}^3}+\text{b}^2\text{c}+\text{b}\text{c}^2-(\text{c}^3+\text{b}^2\text{c}+\text{bc}^2)\times\text{x}^{\text{c}^3}$
$+\text{c}^2\text{a}+\text{ca}^2-(\text{a}^3+\text{c}^2\text{a}+\text{ca}^2)$
$=\text{x}^{\text{a}^3}-\text{b}^3\times\text{x}^{\text{b}^3-\text{c}^3}\times\text{x}^{}\text{c}^3-\text{a}^3$
$=\text{x}^{\text{a}^3}-\text{b}^3+\text{b}^3-\text{c}^3-\text{a}^3$
$=\text{x}^0$
$=1$
Or, Therefore, $LHS = RHS$ Hence proved