সরল করো:
\(\cfrac{3\sqrt7}{\sqrt2+\sqrt5}-\cfrac{5\sqrt5}{\sqrt2+\sqrt7}+\cfrac{2\sqrt2}{\sqrt5+\sqrt7}\)
Madhyamik 2006 , 2003
\(\cfrac{3√7}{√5+√2}-\cfrac{5√5}{√2+√7}+\cfrac{2√2}{√7+√5}\)
\(=\cfrac{3√7}{√5+√2}-\cfrac{5√5}{√7+√2}+\cfrac{2√2}{√7+√5}\)
\(=\cfrac{3√7 (√5-√2)}{(√5+√2)(√5-√2)} \)
\(-\cfrac{5√5 (√7-√2)}{(√7+√2)(√7-√2)}\)
\( +\cfrac{2√2 (√7-√5)}{(√7+√5)(√7-√5)}\)
\(=\cfrac{3√7 (√5-√2)}{(√5)^2 -(√2)^2}-\cfrac{5√5 (√7-√2)}{(√7)^2 -(√2)^2}\)
\(+\cfrac{2√2 (√7-√5)}{(√7)^2 -(√5)^2}\)
\(=\cfrac{3√7 (√5-√2)}{5-2}-\cfrac{5√5 (√7-√2)}{7-2}\)
\(+\cfrac{2√2 (√7-√5)}{7-5}\)
\(=\cfrac{3√7 (√5-√2)}{3}-\cfrac{5√5 (√7-√2)}{5}\)
\(+\cfrac{2√2 (√7-√5)}{2}\)
\(=√7 (√5-√2)-√5 (√7-√2)\)
\(+√2 (√7-√5)\)
\(=√35-√14-√35+√10+√14-√10\)
\(=0\)