How do I get an answer of $14$ using simpsons rule for $\frac{152e}{180n^4}<.0001$

Question

I must have the algebra wrong somewhere but here is the original equation:

$$\frac{152e}{180n^4}<.0001$$

If I then multiply like this:

$$152e<.0001(180)n^4$$

Which then gives:

$$152e < .0018n^4$$

And then dividing:

$$\frac{152e}{.0018} < n^4$$

And then taking the fourth root:

$$\sqrt[4]{\frac{152e}{.0018}}< \sqrt[4]{n^4}$$

And I get about $n = 46.338$

But the book gives a solution of $14$. Can someone explain where I went wrong?

Answer 1

$\dfrac{152 \mathrm{e}}{180n^4}<0.0001$

$152 \mathrm{e} < 0.018n^4$

$\dfrac{152 \mathrm{e} }{0.018} < n^4$

$n > \sqrt[4]{22954.38}$

$n > 12.31$

This still doesn't explain why the solution is listed as $14$, though.