How many digits are after the decimal in $\dfrac{12345678910}{2^{36}\cdot 5^6}$?

[anemone]

Find the number of digits to the right of the decimal point needed to express the fraction $\dfrac{12345678910}{2^{36}\cdot 5^6}$ as a decimal.

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[anemone]

Congratulations to the following members for their correct solutions::)

1. MarkFL
2.kaliprasad
3. lfdahl

Solution from MarkFL:

Spoiler

First, let's reduce the fraction to get:

$$\frac{1234567891}{2^{35}\cdot5^5}$$

Next, let's factor out the power of 10:

$$\frac{1234567891}{2^{30}}\times10^{-5}$$

Next, let's rewrite the mantissa as the sum of powers of 2:

$$\left(2^0+2^{-3}+2^{-6}+2^{-7}+2^{-10}+2^{-12}+2^{-13}+2^{-21}+2^{-23}+2^{-24}+2^{-26}+2^{-29}+2^{-30}\right)\times10^{-5}$$

We now see the mantissa has 31 digits, and the radix/exponent will add an additional 4 zeros to the left of it and to the right of the decimal point, for a total of 35 digits to the right of the decimal point.

Alternate solution from lfdahl:

Spoiler

\[\frac{12345678910}{2^{36}\cdot 5^6}=\frac{12345678910\cdot 5^{30}}{10^{36}}=\frac{1234567891\cdot 5^{30}}{10^{35}}\]

The last digit in the very large nominator will be $5$ (and not $0$). Multiplying this large number by $10^{-35}$ determines the number of digits to the right of the decimal point, namely $35$.