# The Mathematical Ninja and the Cube Root of 81

“I would have to assume the teacher means $\sqrt[4]{81}$ instead.”

“That’s as may be. But $4\ln(3)$ is 4.4 (less one part in 800). A third of that is $1.4\dot 6$, less one part in 800, call it 1.465.”

“So you’d do $e$ to the power of that?”

“Indeed! $\ln(4)$ is 1.4 less 1%, so 1.386 or so. We’re left with 0.079, and that’s about 8%. So 4.32 is going to be a pretty good estimate.”

“The calculator I definitely didn’t just look at says 4.326.”

“Fine. Then I shall note that $13^3$ is 2197 and $3^7$ is 2187.”

“… huh?”

“So! $\left(\frac{13}{3}\right)^3$ is a tiny smidge more than 81.”

“… ok…?”

“Adjusting… $\left( \frac{13}{3} - x\right)^3 \approx \frac{2197}{27} - \frac{169}{3}x$, so $\frac{169}{3}x = \frac{10}{27}$.”

“Sort of with you.”

“That gives $x = \frac{10}{1521}$, call it $\frac{2}{300}$, and the cube root of 81 is about $4.32\dot 6$.”

“Still a bit of a stretch for a 12 year old, don’t you think?”

“I’ll show you a bit of a stretch,” said The Mathematical Ninja.

* Edited 2020-11-16 to correct a number. Thanks to @htfb for the correction! (I also fixed the category.)