If there are fewer than 8 zeroes between the decimal point and the first nonzero digit in the decimal expansion of (t/10

BTGModeratorVI wrote: ↑
Sun Sep 27, 2020 7:16 am
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

Answer: A
Source: Official guide

First: (t/1000)^4 = (t^4)/(1000^4)
Now recognize that 1000^4 = (10^3)^4 = 10^12
So, (t/1000)^4 = (t^4)/(1000^4) = (t^4)/(10^12)

IMPORTANT: When we divide a number by 10^12, we must move the decimal point 12 spaces to the left
So, for example, 1234567/10^12 = 0.000001234567
Likewise, 8888/10^12 = 0.000000008888
And, 66666666666666/10^12 = 66.666666666666

Now let's check each option

I. 3
It t = 3, then (t^4)/(10^12) = (3^4)/(10^12)
= 81/(10^12)
= 0.000000000081
There are 10 zeroes between the decimal point and the first nonzero digit
Since the question tells us that there are fewer than 8 zeroes between the decimal point and the first nonzero digit, we can ELIMINATE statement I

II. 5
It t = 5, then (t^4)/(10^12) = (5^4)/(10^12)
= 625/(10^12)
= 0.000000000625
There are 9 zeroes between the decimal point and the first nonzero digit
Since the question tells us that there are fewer than 8 zeroes between the decimal point and the first nonzero digit, we can ELIMINATE statement II

III. 9
It t = 9, then (t^4)/(10^12) = (9^4)/(10^12)
= 6561/(10^12)
= 0.000000006561
There are 8 zeroes between the decimal point and the first nonzero digit
Since the question tells us that there are fewer than 8 zeroes between the decimal point and the first nonzero digit, we can ELIMINATE statement III

Answer: A

Cheers,
Brent

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