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Let x = 2001^1002 -2001^-1002 and y = 2001^1002 + 2001^-1002. Find x^2 - y^2.

 May 3, 2021
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Let

\(x = 2001^{1002} -2001^{-1002}\) and

\(y = 2001^{1002} + 2001^{-1002}\).

Find \(x^2 - y^2\).

 

\(\text{Let $a =2001^{1002}$} \\ \text{Let $b =2001^{-1002}$} \\ \text{Let $ab \\\qquad~=2001^{1002} *2001^{-1002}\\\qquad~=2001^{1002-1002}\\\qquad~=2001^0\\\qquad~=1 $} \\ \text{Let $x =a-b$} \\ \text{Let $y =a+b$}\)

 

\(\begin{array}{|rcll|} \hline x^2-y^2 &=& (a-b)^2-(a+b)^2 \\ x^2-y^2 &=& a^2-2ab+b^2-(a^2+2ab+b^2) \\ x^2-y^2 &=& a^2-2ab+b^2-a^2-2ab-b^2 \\ x^2-y^2 &=& -2ab-2ab \\ x^2-y^2 &=& -4ab \quad | \quad \mathbf{ab=1} \\ x^2-y^2 &=& -4*1 \\ \mathbf{x^2-y^2} &=& \mathbf{-4} \\ \hline \end{array}\)

 

laugh

 May 3, 2021
edited by heureka  May 3, 2021

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