# Figuring out roots of horrible numbers

“$1296$?!” said the student. “They want me to find the *fourth root* of $1296$?”

“Evidently,” I said.

The air turned, for a moment, blue.

“Well, how about factorising it?”

A different shade of blue. A whirring of pencil. A mutter of 648, a grumble of 324, a harrumph of 162, a yelp of 81. “That would be $2\times2\times2\times2\times3\times3\times3\times3$.”

“How about in powers?”

“$2^4 \times 3^4$,” said the student. “So the fourth root… oh, that’s really annoying. Six.”

“Yup,” I said. “I mean, I know that $1296 = 36^2$ because otherwise I’d have to hand back my dice-game-geek card. But you can always find roots by factorising and dividing all of the powers by the number of the root.”

“So… another example?”

“How about the cube root of $1,728$?”

“I wish I hadn’t asked.” More whirring. More mutters. “OK, got it. $2^6 \times 3^3$, so the cube root is $2^2 \times 3 = 12$.”

“You got it,” I said.

“When does the Mathematical Ninja get back?” asked the student. “I’m bored.”