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 a population of 5 million to 40 million over the course on an hour, by what factor does the population increase every 10 minutes?
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From Manhattan GMAT book, the final solution step before the answer is as follows. I don't question the method or answer itself. I simply don't get how 20th root of 2 to the power of 10 makes square root of 2. Can anyone explain?
How 20th root of 2 to the 10th power makes this?
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In 60 minutes the population increases from 5 million to 40 million.yumi2012 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 a population of 5 million to 40 million over the course on an hour, by what factor does the population increase every 10 minutes?
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From Manhattan GMAT book, the final solution step before the answer is as follows. I don't question the method or answer itself. I simply don't get how 20th root of 2 to the power of 10 makes square root of 2. Can anyone explain?
So r^60 = 40/5
r^60 = 8
(r^20)^3 = 2^3
r^20 = 2
To find the factor by which the population increases in 10 minutes, we have to find r^10
r^10 = sqrt(2)
To answer your question,
20th root of 2 = 2^(1/20) = p
p to the power of 10 = p^10 = (2^(1/20))^10 = 2^(10/20) = 2^(1/2) = sqrt(2)
Take a simple example : square root of 9 to the power of 4 = (sqrt(9))^4 = 3^4 = 81
This can also be written as : (9^(1/2))^4 = 9^(4/2) = 9^2 = 81
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We can plug in the answer choices, which the GMAT would provide.yumi2012 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 a population of 5 million to 40 million over the course on an hour, by what factor does the population increase every 10 minutes?
√2
√3
2
3
4
The answer choices represent the factor by which the population is multiplied every 10 minutes.
Over the course of 60 minutes, there will be SIX 10-minute periods.
Thus, the population will be multiplied by the correct factor SIX TIMES.
When the correct factor is plugged in, the population will increase from 5 million to 40 million.
Answer choice C: 2
5 * (2*2*2*2*2*2) = 320.
Here, the population increases too much.
Eliminate C, D and E.
Answer choice B: √3
5 * (√3 * √3 * √3 * √3 *√3 *√3) = 135.
Here, the population increases too much.
Eliminate B.
The correct answer is A.
Answer choice A: √2
5 * (√2 * √2 * √2 * √2 *√2 *√2) = 40.
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There are two important exponent laws you need to know for this question, and for the GMAT in general.yumi2012 wrote: From Manhattan GMAT book, the final solution step before the answer is as follows. I don't question the method or answer itself. I simply don't get how 20th root of 2 to the power of 10 makes square root of 2. Can anyone explain?
nth root of x = x^(1/n)
Examples:
fifth root of b = b^(1/5)
cube root of k = k^(1/3)
square root of c = c^(1/2)
20th root of 2 = 2^(1/20)
(x^a)^b = x^ab
Examples:
(x^3)^5 = x^15
(k^2)^9 = k^18
(q^10)^6 = q^60
[2^(1/20)]^10 = 2^(1/2)
So, (20th root of 2)^10 = [2^(1/20)]^10 [using the blue rule]
= 2^(1/2) [using the green rule]
= sqrt2 [using the blue rule]
Cheers,
Brent