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Simplify \(\frac{\left(6^x\cdot 12^{x+1}\cdot 20^{x+2}\cdot 30^{x+3}\cdot 42^{x+4}\cdot 56^{x+5}\cdot 72^{x+6}\cdot 90^{x+7}\right)}{10!\cdot 8^x\cdot 15^{x+1}\cdot 24^{x+2}\cdot 35^{x+3}\cdot 48^{x+4}\cdot 63^{x+5}\cdot 80^{x+6}}\)

 Oct 17, 2019
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Simplify

\(\dfrac{6^x\cdot 12^{x+1}\cdot 20^{x+2}\cdot 30^{x+3}\cdot 42^{x+4}\cdot 56^{x+5}\cdot 72^{x+6}\cdot 90^{x+7} }{10!\cdot 8^x\cdot 15^{x+1}\cdot 24^{x+2}\cdot 35^{x+3}\cdot 48^{x+4}\cdot 63^{x+5}\cdot 80^{x+6}} \)

 

\(\begin{array}{|rcll|} \hline && \mathbf{ \dfrac{6^x\cdot 12^{x+1}\cdot 20^{x+2}\cdot 30^{x+3}\cdot 42^{x+4}\cdot 56^{x+5}\cdot 72^{x+6}\cdot 90^{x+7} }{10!\cdot 8^x\cdot 15^{x+1}\cdot 24^{x+2}\cdot 35^{x+3}\cdot 48^{x+4}\cdot 63^{x+5}\cdot 80^{x+6}} } \\\\ &=& \dfrac{(2\cdot3)^x\cdot (3\cdot4)^{x+1}\cdot (4\cdot5)^{x+2}\cdot (5\cdot6)^{x+3}\cdot (6\cdot7)^{x+4}\cdot (7\cdot8)^{x+5}\cdot (8\cdot9)^{x+6}\cdot (9\cdot10)^{x+7} } {10!\cdot (2\cdot4)^x\cdot (3\cdot5)^{x+1}\cdot (4\cdot6)^{x+2}\cdot (5\cdot7)^{x+3}\cdot (6\cdot8)^{x+4}\cdot (7\cdot9)^{x+5}\cdot (8\cdot10)^{x+6}} \\\\ &=& \dfrac{2^x3^x3^{x+1}4^{x+1}4^{x+2}5^{x+2}5^{x+3}6^{x+3}6^{x+4}7^{x+4}7^{x+5}8^{x+5}8^{x+6}9^{x+6}9^{x+7}10^{x+7} } {10!2^x4^x3^{x+1}5^{x+1}4^{x+2}6^{x+2}5^{x+3}7^{x+3}6^{x+4}8^{x+4}7^{x+5}9^{x+5}8^{x+6}10^{x+6}} \\\\ &=& \dfrac{3^x4^{x+1}5^{x+2}6^{x+3}7^{x+4}8^{x+5}9^{x+6}9^{x+7}10^{x+7} } {10!4^x5^{x+1}6^{x+2}7^{x+3}8^{x+4}9^{x+5}10^{x+6}} \\\\ &=& 3^x9^{x+7} \dfrac{4^{x+1}5^{x+2}6^{x+3}7^{x+4}8^{x+5}9^{x+6}10^{x+7} }{10!4^x5^{x+1}6^{x+2}7^{x+3}8^{x+4}9^{x+5}10^{x+6}} \\\\ &=& 3^x9^{x+7} \dfrac{4^{x+1-x}5^{x+2-x-1}6^{x+3-x-2}7^{x+4-x-3}8^{x+5-x-4}9^{x+6-x-5}10^{x+7-x-6} }{10!} \\\\ &=& 3^x9^{x+7} \dfrac{4\cdot5\cdot6\cdot7\cdot8\cdot9\cdot10 }{10!} \\\\ &=& 3^x9^{x+7} \dfrac{4\cdot5\cdot6\cdot7\cdot8\cdot9\cdot10 }{3!4\cdot5\cdot6\cdot7\cdot8\cdot9\cdot10} \\\\ &=& \dfrac{3^x9^{x+7}} {3!} \\\\ &=& \dfrac{3^x9^{x+7}} {2\cdot 3} \\\\ &=& \dfrac{1}{2}\cdot 3^{x-1}9^{x+7} \\\\ &=& \dfrac{1}{2}\cdot 3^{x-1}3^{2(x+7)} \\\\ &=& \dfrac{1}{2}\cdot 3^{x-1+2(x+7)} \\\\ &=& \mathbf{\dfrac{1}{2}\cdot 3^{3x+13}} \\ \hline \end{array}\)

 

laugh

 Oct 18, 2019

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