# The Mathematical Ninja and the Cube Root of 13

A physicist.

A calculator.

The Mathematical Ninja’s face - what could be seen of it - was more snarl than feature. It’s quite tricky to hiss something that doesn’t have any sibilant consonants, but they hissed all the same: “The cube root of 13? You don’t need a calculator for that.”

The student, mindful of the previous episode involving an argument about whether Mars was 5cm away because the calculator said so, surreptitiously returned the machine to the bag. “Yes, sensei. Sorry sensei. I suppose it’s between 2 and 3? A bit less than halfway?”

At least the student was showing willing.

“The cube root of 13.5 is $\frac{3}{\sqrt[3]{2}}$, and the cube root of 2 is about $\frac{5}{4}$, so $\frac{12}{5} = 2.4$ is an overestimate.”

The student, wisely, kept the thought “of course, everyone knows the cube root of 2” from flashing across his face.

“You can also do it with logarithms. $\ln(12) \approx 1.38 + 1.10 = 2.48$, and $\ln(13)$ is a twelfth - call it 0.08 - more than that, or 2.56. A third of that is 0.85, give or take.”

Not a mutter from the student.

“And $e^{0.85}$… well, there are several ways to approach that. It’s roughly the geometric mean of 2 and $e$, which I’d approximate with the arithmetic mean and call 2.35 or so. Let’s go with that.”

The student just nodded, and made a note to check it on the calculator later.