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x=10^100, x^x=10^k, k= [#permalink ]
09 Aug 2006, 00:39

x=10^100, x^x=10^k, k=?

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x^x=10^k
=> (10^100)^(10^100) = 10^k
=> 10^(10^102) = 10^k
k= 10^102

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x^x = 10^k
so xlogx = k
so 10^100 log (10^100)
so 10^100 *100 log 10
so 10^102

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k=102
100*10^100=10^k
10^102=10^k
thus k=102

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k = 10^102
x^x = 10^k
x= 10^100
x^x =((10^100)^(10^100))
= (10^(100x(10^100)))
= (10^(10^2x10^100))
= (10^(10^102))
k = 10^102

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10^100 = 10000.......100 zeros.........

x^x = (10^100)^ (1000........100 zeros)

= 10^(1000....102 zeros)

so k = 102

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ps_dahiya wrote:

10^100 = 10000.......100 zeros......... x^x = (10^100)^ (1000........100 zeros) = 10^(1000....102 zeros) so k = 102

Isn't k = (1000....102 zeros) = 10^102?

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haas_mba07 wrote:

ps_dahiya wrote:

10^100 = 10000.......100 zeros......... x^x = (10^100)^ (1000........100 zeros) = 10^(1000....102 zeros) so k = 102

Isn't k = (1000....102 zeros) = 10^102?

Indeed it is 10^102

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x=10^100 or 10^10^2
x^x=10^(10^2)*10^100=10^102=10^K
k=10^102

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Can somone please send me a page where I can learn this technique?
Really, this is the last thing I need to learn
I have tried every single GMAT prep course and nothing has helped:(

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x=10^100
x^x = (10^100)^10^100
= [10^(10^2*10^100)]
= 10^(10^102)
10^k = 10^(10^102)
K = 10^102

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I totally don't get this. Can someone explian this problem SLOWLY?
Thanks!!!!

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Please explain what I'm getting wrong here:
sure 100 * 10^100 would equal 10^102
But this is 100 ^ 10^100
How is this still 10^102?
Am I missing something really simple? Thanks!!!

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nothanks wrote:

Please explain what I'm getting wrong here: sure 100 * 10^100 would equal 10^102 But this is 100 ^ 10^100 How is this still 10^102? Am I missing something really simple? Thanks!!!

x=10^100

we need to find x^x

x = 10^100 = 100000............100 zeros.......... let this be equal to y

100y = 100000.............102 zeros................ = 10^102

Now x^x = x^y

(10^100)^y

= 10^100y

= 10^(10^102)

This is equal to 10^q

so 10^(10^102) = 10^q

hence q = 10^102

Hope this helps

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I just figured it out. In case anyone else is having a tough time (seems like it's only me!) here is how I got it.
get rid of the first 10^ as it is not important for finding the answer.
so it's
100^10^100
We can break that down to
1^100 * 1^10^100
Further...
1^10^2 * 1^10^100
Even Further...
1^10^102
And finally
10^102
I feel kind of foolish for not making the connection sooner. Oh well...

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This problem is easier if you know logs:

x = 10^100

therefore log x = 100 (1)

x^x = 10^k

therefore xlogx = k

substitute from (1) above:

x(100) = k

therefore k = 100.10^100 = 10^2.10^100 = 10^(100+2) = 10^102

haas_mba07 wrote:

ps_dahiya wrote:

10^100 = 10000.......100 zeros......... x^x = (10^100)^ (1000........100 zeros) = 10^(1000....102 zeros) so k = 102

Isn't k = (1000....102 zeros) = 10^102?

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gx^x=(10^100)^(10^100)
= 10^(100^(10^100)
X^x=10^k
Take logs.
Xlogx=klog10=k
10^100(log10^100)=k
10^100(100)=k
10^100*10^2=k
10^102=k

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