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If x = 5^25 and x^x = 5^k, what is k

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If x = 5^25 and x^x = 5^k, what is k [#permalink]

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If x = 5^25 and x^x = 5^k, what is k

A. 5^26
B. 5^27
C. 5^50
D. 5^52
E. 5^625
[Reveal] Spoiler: OA

Last edited by qw1981 on 26 Dec 2014, 04:11, edited 1 time in total.

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Re: If x = 5^25 and x^x = 5^k, what is k [#permalink]

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New post 26 Dec 2014, 03:58
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x=5^25
=> x^x = (5^25)^(5^25 ) = 5^(25 * 5^25 ) [because (x^y)^z = x*yz)]
= 5^(5^2 * 5^25) = 5^(5^(2+25)) [be]cause x^a * x^b = x^(a+b)
= 5^(5^27)

Given that
x^x=5^k
=> 5^(5^27) = 5^k
Since the base is same so the power will also be same
=> k = 5^27

So, answer will be B

Hope it helps!
qw1981 wrote:
IF x=5^25 and x^x=5^k, what is k

a)5^26 b)5^27 c)5^50 d)5^52 e) 5^625


PS: I cannot understand the solution that is available for this problem. I solved it another way which was tad bit longer than what was used to explain this question.

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Re: If x = 5^25 and x^x = 5^k, what is k [#permalink]

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New post 26 Dec 2014, 04:12
F x=5^25 and x^x=5^k, what is k

a)5^26 b)5^27 c)5^50 d)5^52 e) 5^625


Since x=5^25
x^x=[5^25]^(5^25)
=5^[(5^2)*(5^25)]
=5^(5^27)

Answer B
I am not able type the power properly if someone could help with this.

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Re: If x = 5^25 and x^x = 5^k, what is k [#permalink]

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New post 26 Dec 2014, 07:46

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Exponents problem [#permalink]

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See the file attached. Does anyone know how we pass from 5^25 to 25 ?
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Screen Shot 2015-09-11 at 6.24.14 AM.png
Screen Shot 2015-09-11 at 6.24.14 AM.png [ 229.12 KiB | Viewed 2054 times ]

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Re: Exponents problem [#permalink]

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New post 11 Sep 2015, 11:22
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Christophestanic wrote:
See the file attached. Does anyone know how we pass from 5^25 to 25 ?

Dear Christophestanic,
I'm happy to respond. :-)

I gather that you are new to GMAT Club. First, I will give you some advice.
1) This particular subforum, the "Ask GMAT Expert" forum, is for asking general study questions (study plans, book reviews, etc.). For a particular math or verbal content question, you should post in the math or verbal forums, not here.
2) VERY IMPORTANT: Never start a brand new thread for a math question before searching for it. Any GMAT practice math question from virtually any source has already been posted here and discussed in depth. It adds unnecessarily to the complexity of this site to start a brand new thread on a question that has already been discussed. More importantly, you probably can find your answer already here, in the discussions in these pre-existent threads. For example, I found this math question here:
if-x-5-25-and-x-x-5-k-what-is-k-190765.html

Please keep all that in mind when you post in the future.

Now, as to your math question: what's a little embarrassing for Veritas is that the answer they give on that screenshot is not correct. The variable k does not equal 27 ---- rather it equals k = 5^27. Actually, there are a few mistakes on that screenshot. Usually, Veritas is a little more solid than this.

Rather than explain their version, which has multiple mistakes, let's just start from scratch.

x = 5^25, a big number

x^x = (5^25)^(5^25), a super-big number

To simplify this, I am going to call the first 25 A, and the final part in parentheses B, so A = 25, and B = (5^25)

Then,
x^x = (5^A)^B

In this form, you should recognize it as one of the laws of exponents: a power to an exponent means multiply the exponents.
x^x = (5^A)^B = 5^(AB) = 5^k

Thus, k is simply the product of what I called A and B.

K = A*B = (25)*(5^25) = (5^2)*(5^25) = 5^(2 + 25) = 5^27

Notice we used the "product of the powers" law of exponents there. Once again, notice that k does NOT equal 27, as claimed on that screenshot. Instead, k = 5^27.

BTW, you don't need to know this, but some cool big number facts:
k = 5^27 = 7.45 x 10^18, more than a billion squared.

x^x =5^k would be a number with more than 5.2 x 10^18 digits. We would need a piece of paper larger than our entire galaxy simply to write this number down!

Here are some more challenging exponent problems for practice:
http://magoosh.com/gmat/2014/challengin ... and-roots/

Does all this make sense?
Mike :-)
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Exponents problem [#permalink]

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New post 11 Sep 2015, 12:54
Christophestanic wrote:
See the file attached. Does anyone know how we pass from 5^25 to 25 ?


mikemcgarry, thank you for talking about the guidelines for posting.

Christophestanic, please follow posting guidelines (rules-for-posting-please-read-this-before-posting-133935.html).

Also, do not use pictures to stand for actual questions.

As per the guidelines, type in the complete question with 5 options and OA for the same. This is very important for initiating a discussion on that particular question.

Moved to PS forum.

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Re: If x = 5^25 and x^x = 5^k, what is k [#permalink]

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New post 30 Sep 2017, 21:43
qw1981 wrote:
If x = 5^25 and x^x = 5^k, what is k

A. 5^26
B. 5^27
C. 5^50
D. 5^52
E. 5^625


x = 5^25
x^x = (5^25)^(5^25)
= (5^(5^2))^(5^25)
= (5)^(5^2 * 5^25)
= 5^(5^27)

Answer B
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Re: If x = 5^25 and x^x = 5^k, what is k   [#permalink] 30 Sep 2017, 21:43
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