“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.)
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