Hi,

Since the population multiplies every 10 minutes, the population needs to multiply 6 times to complete the hour.

Assume factor is x

5*x^6=40

[spoiler]Therefore x = root 2[/spoiler]

Troika wrote:A strain of bacteria multiplies such that the ratio of its population in any two consecutive minutes is constant. If the bacteria grows from a population of 5 million to a population of 40 million in one hour, by what factor does the population increase every 10 minutes?

The easiest approach would be to plug in the answer choices, which an actual GMAT question would provide.
The answer choices would represent the factor by the population increases every 10 minutes.
Over 60 minutes, the population will be multiplied by the given factor 60/10 = 6 times.

Answer choice: √2.
5,000,000 * (√2)^6 = 5,000,000 * 8 = 40,000,000.

The correct answer is [spoiler]√2[/spoiler].

We also could use the formula for exponential growth:

Final Amount = Original Amount * (Multiplier)^(Number of changes)

In the problem above:
Final Amount = 40,000,000
Original Amount = 5,000,000
Multiplier = x
Number of changes = 6 (We're looking for the change over 10 minutes, and this change will occur 6 times over the course of 60 minutes.)

Plugging these values into the formula:
40,000,000 = 5,000,000 * x^6
8 = x^6
x = 8^(1/6) = √2.

Thanks Anup and GMATGuruNY!