# Find the smallest positive integer n

#### anemone

##### MHB POTW Director
Staff member
Find the smallest positive integer $n$ for which $n^{16}$ exceeds $16^{18}$.

##### Well-known member
Find the smallest positive integer $n$ for which $n^{16}$ exceeds $16^{18}$.
23 as below

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

##### Well-known member
Hello, anemone!

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

I used a different approach.

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: . $$n \;=\;23$$

##### Well-known member
Hello, anemone!

I used a different approach.

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