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How 20th root of 2 to the 10th power makes this?

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How 20th root of 2 to the 10th power makes this?

by yumi2012 » Mon Aug 12, 2013 8:21 pm
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?
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by ganeshrkamath » Tue Aug 13, 2013 12:04 am
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?

---------

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?
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In 60 minutes the population increases from 5 million to 40 million.
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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by GMATGuruNY » Tue Aug 13, 2013 2:36 am
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
We can plug in the answer choices, which the GMAT would provide.
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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by Brent@GMATPrepNow » Tue Aug 13, 2013 6:07 am
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?
There are two important exponent laws you need to know for this question, and for the GMAT in general.

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
Brent Hanneson - Creator of GMATPrepNow.com
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