TwiddleLab / Math / Growth: Exponents and Logs
Half Steps: Roots as Powers
What you will learn
- A half step of doubling multiplies by the number that, done twice, gives 2: 2^(1/2) = √2 ≈ 1.414.
- A fraction exponent is a root: a^(1/2) = √a, a^(1/3) is the cube root of a.
- a^(p/q) is the q-th root, done p times: 8^(2/3) = 2² = 4.
How the lesson goes
A colony doubles every hour. How much bigger is it after half an hour?
Drag the lever to your bet.
Find the half-hour multiplier.
Every half hour the colony is multiplied by the same number. Two half hours must make exactly ×2.
Two equal half steps must multiply to the full step: m × m = 2. The number whose square is 2 is √2 ≈ 1.414.
Run the clock with your multiplier.
Press Play and watch the table at every whole hour.
Half a step of ×2 is ×2^(1/2).
2^(1/2) = √2 ≈ 1.414, because 2^(1/2) × 2^(1/2) = 2^(1/2 + 1/2) = 2¹.
A third of a step is the cube root: 2^(1/3) × 2^(1/3) × 2^(1/3) = 2.
In general a^(p/q): take the q-th root of a, then raise it to p.
Now split the hour into three.
Every 20 minutes the colony is multiplied by the same number. Three steps must make exactly ×2.
Three equal steps must multiply to 2, so m³ = 2 and m is the cube root of 2, written 2^(1/3) ≈ 1.26.
Game: root match.
Tap the value of each card. The picture explains each one. 60 seconds.
A piano octave doubles the frequency.
From A = 220 Hz to the next A = 440 Hz are 12 equal steps. Tap keys to hear them, then pick the key exactly half an octave up.
Each of the 12 steps multiplies by the same number, so 12 steps give ×2. Six steps give the half-way multiplier 2^(6/12) = √2, and 220 × √2 ≈ 311 Hz.
When is the colony 2.83 times bigger?
It doubles every hour. Drag the clock lever to the time, then Check.
Exponents add when you multiply: 2^(3/2) = 2^1 × 2^(1/2). That is a full doubling and a half doubling: 2 × 1.414 ≈ 2.83.
One more trick: run the clock backwards.
The colony has 100 now. How many were there half an hour ago?
Previous lesson: Shrinking: Zero and Negative Powers · Next lesson: Growth Curves and Percent Growth · Part of the course: Growth: Exponents and Logs
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