MHB Find the smallest positive integer n

[anemone]

Find the smallest positive integer $n $ for which $n^{16}$ exceeds $16^{18}$.

 

[kaliprasad]

Find the smallest positive integer $n $ for which $n^{16}$ exceeds $16^{18}$.

23 as below

Spoiler

n^16 > 16^18
or n^8 > 16^9 or 4^18
or n^4 > 4^9 or 2^18
or n^2 > 2^9 or 512
n = 22 => n^2 = 484 and n = 23 => n^2 = 529

 

[soroban]

Hello, anemone!

\text{Find the smallest positive integer }n
. . \text{ for which }n^{16}\text{ exceeds }16^{18}.

kaliprasad is correct.
I used a different approach.

Spoiler

We want: .n^{16} \;> \; 16^{18}

. . . . . . . .n^{16} \;>\; (2^4)^{18}

. . . . . . . .n^{16} \;>\;2^{72}

. . . . . . . . . n \;>\;2^{\frac{72}{16}}\;=\;2^{\frac{9}{2}}

. . . . . . . . . n \;>\; 2^{4+\frac{1}{2}} \;=\;2^4 \cdot 2^{\frac{1}{2}}

. . . . . . . . . n \;>\; 16\sqrt{2} \;=\; 22.627417

Therefore: . n \;=\;23

 

[kaliprasad]

Hello, anemone!


kaliprasad is correct.
I used a different approach.

Spoiler

We want: .n^{16} \;> \; 16^{18}

. . . . . . . .n^{16} \;>\; (2^4)^{18}

. . . . . . . .n^{16} \;>\;2^{72}

. . . . . . . . . n \;>\;2^{\frac{72}{16}}\;=\;2^{\frac{9}{2}}

. . . . . . . . . n \;>\; 2^{4+\frac{1}{2}} \;=\;2^4 \cdot 2^{\frac{1}{2}}

. . . . . . . . . n \;>\; 16\sqrt{2} \;=\; 22.627417

Therefore: . n \;=\;23

above approach is more straight forward