\((10^4 * 3.456789)^8=34567.89^8=(3456789*10^{-2})^8=3456789^8*10^{-16}\).

Answer: C.

Hope it's clear.
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First of all split the terms (10^4)^8 * (3.456789)^8 --------> (10)^32 * (3.456789)^8

10^32 will have 32 zeros after 1.

(3.456789)^8 ----> Here we should understand 2 things 1) Any decimal when squared its decimal places doubles e.g. 1.2^2 = 1.44 2) x^2 means (x^2)^2)^2

So here when 3.456789 squared its decimal places would become 12. Squaring the resultant figure again would give 24 decimal places AND squaring the resultant figure for the final time would give 48 decimal places.

So we will have the figure as (x.abcd....48 times)*(1000000.....32 times). The 32 zeros would shift the decimal sign of x.abcd... to the right side by 32 places. So beyond decimal point we will have 48 - 32 = 16 places. Hence C
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(34567.89)^8
Whenever you multiply a decimal by itself, you get the number of decimals times the power for the numbers on the right
0.2^2 = 0.04, 0.2^3=0.008
So, 2 numbers to the right of the decimal times power 8.
This is how I got the answer but if there's any flaw in the logic please let me know.

Well, what I did was to do the multiplication in the brackets and end up with this: 3,4567.89

Now, we want to raise 3,4567.89 to the eighth power. Instead of doing this, I tried with 0.01. I noticed that 0.01^2, is adding two more digits: 0.0001.

So, ading 2 more digits until we reach the eighth power will lead to 14 digits. Plus the 2 we already have (.89) sums up to 16 digits. ANS C

Since 10^4 x 3.456789 = 34567.89, (34567.89)^8 will have a total of 2 x 8 = 16 digits to the right of the decimal place.

Answer: C
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