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.)

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

Share

This site uses Akismet to reduce spam. Learn how your comment data is processed.