Determine largest number:

**great_math** · October 28th 2008, 12:19 AM

$\ 1,\sqrt{2},\sqrt[3]{3}.....,\sqrt[n]{n}$

Determine the largest number in the above sequence

**mr fantastic** · October 28th 2008, 05:02 AM

Originally Posted by great_math

$\ 1,\sqrt{2},\sqrt[3]{3}.....,\sqrt[n]{n}$

Determine the largest number in the above sequence

The function $y = x^{1/x}$ has a maximum turning point at x = e. Since 2 < e < 3, the question now becomes a simpler one:

"Which is bigger, $\sqrt{2}$ or $\sqrt[3]{3}$ " .... Which is more or less the same as asking "Which is bigger, $(\sqrt{2})^6$ or $(\sqrt[3]{3})^6$ " ....

Alternatively, note that $\sqrt{2}$ has the same value as $\sqrt[4]{4} = 4^{1/4}$ .....

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