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 #3
avatar+26364 
+7

the forth root of 40000? $$\sqrt[4]{40000}=\;?$$

 

$$\sqrt[4]{40000}=$$

$$\\=\sqrt[2]{\sqrt[2]{40000}}\qquad 40000 = 4 * 100 * 100=2*2*100*100=2^2*100^2\\\\
=\sqrt[2]{\sqrt[2]{2^2*100^2}}\\\\
=\sqrt[2]{\sqrt[2]{2^2}\times\sqrt[2]{100^2}}\\\\
=\sqrt[2]{2*100}\qquad 2*100= 2 * 10 * 10=2*10^2 \\\\
=\sqrt[2]{2*10^2}\\\\
=\sqrt[2]{2}\times\sqrt[2]{10^2}}\\\\
=\sqrt2*10\\\\
=10\sqrt2$$

Latex code:

 \\=\sqrt[2]{\sqrt[2]{40000}}\qquad 40000 = 4 * 100 * 100=2*2*100*100=2^2*100^2\\\\
=\sqrt[2]{\sqrt[2]{2^2*100^2}}\\\\
=\sqrt[2]{\sqrt[2]{2^2}\times\sqrt[2]{100^2}}\\\\
=\sqrt[2]{2*100}\qquad 2*100= 2 * 10 * 10=2*10^2
=\sqrt[2]{2*10^2}\\\\
=\sqrt[2]{2}\times\sqrt[2]{10^2}}\\\\
=\sqrt2*10\\\\ 
=10\sqrt2

$$\sqrt[4]{40000}=$$ $${\mathtt{10}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{2}}}} = {\mathtt{14.142\: \!135\: \!623\: \!730\: \!950\: \!5}}$$

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May 21, 2014