Mathematics · Powers and exponential growth

Powers and exponential growth

On which day is a pond half covered by water lilies that double every day? Learn about powers, the rules of exponents, doubling, how linear, exponential and logistic growth differ, and the rule of 70 for doubling time.

25 min
1 simulation
Mini quiz

Updated: 24 October 2026

Preview

The question

There is a single water lily leaf in a pond. The number of leaves doubles every day, and at the end of day 30 the surface of the pond is completely covered. On which day was the pond half covered? Is your first answer day 15?

Here is another old puzzle: 1 grain of rice is placed on the first square of a chessboard, 2 on the second, 4 on the third; each square gets twice as many as the one before. A chessboard has 64 squares. How many grains go on the last square?

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Number line

Fold the paper

Drag the point or use the buttons to move in steps of 1 fold. When you reach the right value, press Check.

Predict first

About how many folds does it take to make the thickness 100 times greater?

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Lab

Exponential growth: the lily pond

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Fig. 1The lily pond

Move the day forward or press play: three ponds start the same and grow by different rules

day 7 · Exponential: 12.5%

Day tday 7

Daily factor RR = 2

Your own values

R (factor)

Daily growth

Start N₀

Length (days)

Day t

R from 1.001 to 10 (or type the daily growth as a percentage), at most 120 days, at most 100 leaves to start. A value applies when you press Enter or leave the box.

Today: day 7

T₂ = 1 days

Linearsame amount each day	8	0.781%
Exponentialsame proportion each day	128	12.5%
Logisticslows as the pond fills	114	11.1%

Daily growth: 100%
Day the exponential pond fills: 10
Day it is half full: 9
Day the logistic pond is half full: 10

A pond holds 1,024 leaves (2¹⁰ = 32 × 32).

Leaves – days

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1Predict
2Try
3Explain

The leaves double every day and the pond is completely covered on day 10. On which day was it half covered?

In exponential growth each day adds the same proportion, not the same amount: N = N₀ × Rᵗ. That is why for a long time it looks as if nothing is happening, and then a few doublings change everything; one doubling time before the pond is full (one day when R = 2) it is still half empty. Nothing can grow exponentially for ever: when space, food or resources run out, growth slows. The logistic curve is the simplest model of that. This is an abstract model, not a forecast for a real pond, disease or population. See A. A. Bartlett, “Forgotten Fundamentals of the Energy Crisis”, Am. J. Phys. 46, 876 (1978; his lecture is titled “Arithmetic, Population and Energy”); Verhulst’s logistic equation (1838).

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Frequently asked

5 questions

Why is 2 to the power of 10 not 20?

You do not multiply the base by the exponent: aⁿ is the product of n copies of a. 2 × 10 = 20, but 2¹⁰ is ten 2s multiplied together: 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1024.

Why is any number to the power of zero equal to 1?

2³ = 8, 2² = 4, 2¹ = 2: each time the exponent goes down by one, the result is halved. If the pattern continues, 2⁰ = 1. The division rule says the same: 2³ ÷ 2³ = 1, and subtracting the exponents gives 2³⁻³ = 2⁰. So a⁰ = 1 for every a that is not zero.

When do I add exponents and when do I multiply them?

When you multiply two powers with the same base, you add the exponents: 2³ · 2⁴ = 2⁷. When you raise a power to a power, you multiply the exponents: (2³)² = 2⁶. If the bases are different (2³ · 3²), these rules do not apply; work out each power first: 8 · 9 = 72.

What is the difference between linear and exponential growth?

In linear growth the same amount is added at each step: 1, 2, 3, 4… In exponential growth you multiply by the same number at each step: 1, 2, 4, 8… At first the two are close, but exponential growth soon races ahead: after 20 steps the linear one reaches 21 and the doubling one reaches 1,048,576.

Does the rule of 70 always give the right answer?

No, it is an approximation. It gives the doubling time of something that grows by p% each period as about 70 ÷ p. It works well for small percentages: for 7% the rule says 10 periods and the true value is about 10.2. The error grows with the percentage: at 100% growth the amount doubles in 1 period, but the rule says 0.7.