A bacterium multiplies by splitting in two. It copies its DNA, grows, and divides into two new cells. Then each of those divides. This is a place to use exponents.
Key idea
With food and warmth, the number of bacteria doubles each generation: 1, 2, 4, 8, 16 … After $n$ generations, one bacterium becomes $2^n$; a starting number $N$ becomes $N \times 2^n$. Growth slows and stops when food and space run out.
Why it works
Every bacterium divides, the new ones included, so the number is multiplied by 2 each time, not increased by 2. That is why the counts take off. In ideal lab conditions E. coli can divide about every 20 minutes, so one cell could become $2^{3} = 8$ in an hour and $2^{30}$, over a billion, in ten hours, if nothing ran out.
Worked examples
Worked example: Count the doublings
One bacterium divides every generation. How many are there after 5 generations?
- $1 \to 2 \to 4 \to 8 \to 16 \to 32$.
- Or: $2^5 = 32$.
Answer: 32 bacteria.
Worked example: Use the time
A sample starts with 10 bacteria that divide every 30 minutes. How many after 2 hours?
- Generations: $120 \div 30 = 4$.
- Number: $10 \times 2^4 = 10 \times 16 = 160$.
Answer: 160 bacteria.
Common mistake
Adding 2 each generation (2, 4, 6, 8) instead of doubling (2, 4, 8, 16), or expecting doubling to go on forever. Check: each generation should be twice the one before. And in any real dish, food runs low and wastes build up, so the curve levels off.