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If there are fewer than 8 zeroes between the decimal point

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If there are fewer than 8 zeroes between the decimal point

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If there are fewer than 8 zeroes between the decimal point and the first nonzero digit in the decimal expansion of (t/1000)^4, which of the following numbers could be the value of t?

I. 3
II. 5
III. 9

A) None
B) I only
C) II only
D) III only
E) II and III

OA A

Source: Official Guide

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BTGmoderatorDC wrote:
If there are fewer than 8 zeroes between the decimal point and the first nonzero digit in the decimal expansion of (t/1000)^4, which of the following numbers could be the value of t?

I. 3
II. 5
III. 9

A) None
B) I only
C) II only
D) III only
E) II and III

OA A

Source: Official Guide
If t = 1, then (1/1000)^4 = (1/10^3)^4 = 1/10^12. Therefore, the decimal expansion would have 12 decimal places, with the last (rightmost) digit being a 1. That is, there are 11 zeros between the decimal point and the last digit 1. For any of the given t values, 3, 5 and 9, it will not change the number of decimal places; however, it may change the number of zeros between the decimal point and the first nonzero digit.

If t = 3, then t^4 = 3^4 = 81. So 81 will occupy the last two of the 12 decimal places; that means there are 10 zeros between the decimal point and the first nonzero digit 8.

If t = 5, then t^5 = 5^4 = 625. So 625 will occupy the last three of the 12 decimal places; that means there are 9 zeros between the decimal point and the first nonzero digit 6.

If t = 9, then t^5 = 9^4 = 6561. So 6561 will occupy the last four of the 12 decimal places; that means there are 8 zeros between the decimal point and the first nonzero digit 6.

Since we are looking for fewer than 8 zeros between the decimal point and the first nonzero digit, none of the given t values will make this happen.

Answer: A

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Let's try as follows:

\(\left(\frac{t}{1000}\right)^4=\left(\frac{t}{10^3}\right)^4=\frac{t^4}{10^{12}}\)
\(10^{12}\) will move decimal point to the left \(12\) times.

If we test \(9 \Rightarrow 9^4 = 6561\) or \(0,6 \cdot 10^4\)
So \(4-12=8\).

Therefore, the correct answer is __A__

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