Toán Lớp 9: Không dùng máy tính hãy so sánh √2013-√2012 và √2012-√2011
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Toán Lớp 9: Không dùng máy tính hãy so sánh √2013-√2012 và √2012-√2011
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\sqrt {2013} – \sqrt {2012} \\
= \dfrac{{\left( {\sqrt {2013} – \sqrt {2012} } \right).\left( {\sqrt {2013} + \sqrt {2012} } \right)}}{{\sqrt {2013} + \sqrt {2012} }}\\
= \dfrac{{2013 – 2012}}{{\sqrt {2013} + \sqrt {2012} }}\\
= \dfrac{1}{{\sqrt {2013} + \sqrt {2012} }}\\
\sqrt {2012} – \sqrt {2011} \\
= \dfrac{{\left( {\sqrt {2012} – \sqrt {2011} } \right)\left( {\sqrt {2012} + \sqrt {2011} } \right)}}{{\sqrt {2012} + \sqrt {2011} }}\\
= \dfrac{{2012 – 2011}}{{\sqrt {2012} + \sqrt {2011} }}\\
= \dfrac{1}{{\sqrt {2012} + \sqrt {2011} }}\\
Do:\sqrt {2013} + \sqrt {2012} > \sqrt {2012} + \sqrt {2011} \\
\Leftrightarrow \dfrac{1}{{\sqrt {2013} + \sqrt {2012} }} < \dfrac{1}{{\sqrt {2012} + \sqrt {2011} }}\\
\Leftrightarrow \sqrt {2013} – \sqrt {2012} < \sqrt {2012} – \sqrt {2011}
\end{array}$