aⁿ · Exponentiation
A name for repeated multiplication — the exponent.
Fold a 0.1mm sheet of paper 42 times — how far does its thickness reach?
Distinguish the number of factors from the result.
The sage who asked only for one grain of rice — doubled on each chess square.
Open the immersive version →A reading path containing this lesson: Reverse repeated multiplication · 1/3
Why does this question matter?
2 × 2 × 2 × 2 × 2 × 2 × 2… multiplying the same number over and over is everywhere — cell division, interest, data sizes. Writing it out every time is drudgery.
So we agreed to shorten 'multiplied 2 ten times' to 2¹⁰. Just one piece of notation — yet that little superscript becomes the starting point of whole new mathematics: exponent laws, logarithms, exponential functions.
More important is the gut feeling. Repeated multiplication always betrays linear intuition — you have to feel in your bones that 42 folds reach the Moon, or you'll misjudge compound interest, epidemics, and technology.
Pause and try it yourself
Make a prediction first. Change one input at a time and compare what happens with what you expected.
Predict first — fold a 0.1mm sheet 42 times. How far does the thickness reach?
Fold it yourself
0 folds
0.1mm
Current thickness
The fold now = the notation now
Fold once. The notation grows with every fold.
Read the idea from another angle
The exponent records a count, not a size.
The 10 in 2¹⁰ isn't a magnitude — it's how many times you pressed the multiply button. With this one shift of view, all the exponent laws become obvious: counts simply add.
The final fold equals everything that came before it.
If 41 folds reach halfway to the Moon, the single 42nd fold covers the other half. The endgame of exponential growth is always violent.
Now read the definitions with their conditions
Write the pattern you discovered while folding as formulas, and you get the laws of exponents.
Definition of a power
a multiplied n times. n is a count — like the number of times you pressed the fold button.
The law — counts add
Multiply m times, then n more times: (m+n) times in total. Exactly what you counted in the experiment.
Zero and negative exponents
Keep extending 'each step down divides by a': a¹=a, a⁰=1, a⁻¹=1/a. The natural continuation of the pattern.
Make one more distinction
2¹⁰ is probably around 2×10 = 20.
2¹⁰ = 1,024. The exponent is 'how many times you multiplied', not 'what you multiplied by'. Reading repeated multiplication with additive intuition is the most common trap.
2⁰ is 0, and 2⁻¹ is negative.
2⁰ = 1 and 2⁻¹ = 1/2. Extend the pattern 'each step down divides by 2' and both fall out naturally. Not a convention — the extension of a pattern.
Where else can this idea help?
The ladder of data sizes
KB→MB→GB→TB is ×1,024 (=2¹⁰) each step. The world of storage was designed in powers from the start.
Cell division and viruses
1 becomes 2, 2 becomes 4… twenty divisions exceed a million. The basic unit of biological growth is the power.
The skeleton of compound interest
(1+r)ⁿ — the heart of the compounding formula is a power. n sitting in the exponent is the identity of 'time multiplies money'.
Password strength
Each extra character multiplies the possibilities by dozens. Security speaks in powers too.
Explain it in your own words
Try another question
Q1One grain of rice on the first chessboard square, doubling every square — how many grains on square 64? (just estimate the digit count)Show answer ▾
2⁶³ ≈ 9.2 quintillion — a 19-digit number, hundreds of times the world's annual rice production. The legend of the king who promised this reward and went bankrupt exists for a reason.
Q2Using the approximation 2¹⁰ ≈ 1,000, roughly what is 2²⁰?Show answer ▾
2²⁰ = 2¹⁰ × 2¹⁰ ≈ 1,000 × 1,000 = a million. With the law (counts add), estimating huge numbers takes seconds. The exact value: 1,048,576.
Q3Why is 5⁰ equal to 1, not 0? Explain with the pattern.Show answer ▾
5³=125 → 5²=25 → 5¹=5 — each step down divides by 5. The next step is 5÷5=1, so 5⁰=1. The rule, extended naturally.
Try answering yourself before revealing it — getting it wrong is where learning starts.
What question comes next?
A reading path containing this lesson: Reverse repeated multiplication · 1/3
Previous concept
Prime Numbers
Once you know multiplication's atoms (primes), repeatedly multiplying the same atom — the power — is the natural next step.
← Prime Numbers labLeads to next
Logarithms
If the exponent is 'how to write the count of multiplications', the logarithm is 'how to read that count backwards'. Climb the ladder: exponent. Read it down: logarithm.
Go to the Logarithms lab →Related lab