IIT JEE , AIEEE Forum - Mathematics , Algebra

srujana

...find the no.of digits in

₄2

256^{16 divided by......}

^₃

81⁹

nunoxic

Find the no of digits in 256 to the power 16 to the power 4 to the power 2 divided by 27 to the power 9 to the power 3

the numerator will simplify to :
=2 ^ 8 ^ 16 ^ 4 ^ 2
=2 ^ 1024

the denominator will simplify to
=3 ^ 3 ^ 9 ^ 3
=3 ^ 81

let us assume
y = the given sum
log y = 1024 log 2 - 81 log 3
=1024 * 0.3010 - 81 * 0.4771 (these are values u shud know)
=308.224 - 38.6451
=269.5789

thus y will be antilog (269.5789)
log 4 is 0.6021
thus the answer will be around
3. ?? x 10^ 269
thus no of digits must be 270

I have used a very crude way.....
refer to it only if the answer is correct cuz i am not sure whether it can be done this way

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sankydreams

Well its something even divided by a multiple of three. Thus it wud be an irrational term with infinite digits.Well i think so.

umang

Hey nuxoric !

u r wrong at this step -

2 ^ 8 ^ 16 ^ 4 ^ 2
=2 ^ 1024

this is wrong !!!

u hav done 2^8*16*4*2 !!!!!

pls check ur solution !!!

rahulvedanta

wrong question cannot be solved quickly

MUDIT

THE ANSWER SHOULD BE 270 DIGITS
SOLUTION
256 TO THE POWER OF 16 TO THE POWER OF 4 TO THE POWER OF 2 DIVIDED BY 27 TO THE POWER OF 9 TO THE POWER OF 3 CAN BE REDUCED TO 2 THE POWER OF 8x2x4x16 DIVIDED BY 3 TO THE POWER OF 3X3X9.

THIS COMES TO 2 TO THE POWER OF 1024 DIVIDED BY 3 TO THE POWER OF 81.

USING LOGS WE GET 1024log2 - 81log3. TAKING THE VALUES OF log2 AS 0.3010 AND log3 AS 0.4771 WE GET

logx=308.224 - 38.645 WHERE x IS THE REMAINDER
SOLVING THIS WE GET x AS ANTILOG OF 269.579
x WILL THUS HAVE 270 DIGITS

MUDIT

PLEASE RATE IF YOU FOUND THE ANSWER SATISFACTORY.

srujana

Sorry but the question should be...find the no.of digits in

₄2

256^{16 divided by......}

^₃

81⁹

nunoxic

my solution is right i think
wats the answer srujana ??

btw,
[ A ^ m ] ^ n = A ^ mn

srujana

I dont know the answer nunoxic but see ur statement once

[ A ^ m ] ^ n = A ^ mn
the above holds true only when a^m is taken as a whole

i.e.,a to the power m whole to the power n is a power mn.........or 256 to the power 64 whole to the power of 8 is equal to 256 to the power 64*8......

but here we dont take 256 to the power 64 as whole.....

hope u have understood.......................plz try another method

srujana

129 views and still no answer............somebody please answer it.

punterjack

Then why dont u do the problem the way mudit has sugested u.

on taking the log of the expression on gets 1024log 2 - 108 log 3

on taking log 2=.3 and log 3=.4771 as mudit sugested one gets the final log

values as log 251.6732 which means that the number has 252 digits if i amnt

mistaken or if i am wrong u could use the method that mudit sugested.

iamgodofdeath131

The no of digits in

2

4

16

256

divided by

3

9

81

is either 17 or 32

Two answer are bcoz one is excluding the digits after decimal that is 17 and other one is includng digits after decimal that is 32.

if u hav got the correct answer then pls give me rate.I think that i am correct becoz I hav halfly solved it with mathematics mind that is by taking "LOG" and halfly with clanguage mind.

Can u tell me the proper method of it.

a_buddy4uin06

hiiiiiiii.......
this is jus mudits idea.. but i m jus correcting d small mistake he did..

log x = 16¹⁶log256 - 729log81

now if we compute d value...
but i wud like to kno if we can get this ans easily.. cos computing 16¹⁶is time consumin..........

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