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- Feb 14, 2012
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Find the smallest positive integer $n $ for which $n^{16}$ exceeds $16^{18}$.
23 as belowFind the smallest positive integer $n $ for which $n^{16}$ exceeds $16^{18}$.
[tex]\text{Find the smallest positive integer }n[/tex]
. . [tex]\text{ for which }n^{16}\text{ exceeds }16^{18}.[/tex]
above approach is more straight forwardHello, anemone!
kaliprasad is correct.
I used a different approach.
We want: .[tex]n^{16} \;> \; 16^{18}[/tex]
. . . . . . . .[tex]n^{16} \;>\; (2^4)^{18}[/tex]
. . . . . . . .[tex]n^{16} \;>\;2^{72}[/tex]
. . . . . . . . . [tex]n \;>\;2^{\frac{72}{16}}\;=\;2^{\frac{9}{2}}[/tex]
. . . . . . . . . [tex]n \;>\; 2^{4+\frac{1}{2}} \;=\;2^4 \cdot 2^{\frac{1}{2}}[/tex]
. . . . . . . . . [tex]n \;>\; 16\sqrt{2} \;=\; 22.627417[/tex]
Therefore: . [tex]n \;=\;23[/tex]