What is the tenth digit to the right of the decimal point

\((\frac{1}{5})^{10}\)

\(\frac{1}{5^{10}}\)

\(\frac{1}{(\frac{10}{2})^{10}}\)

\(\frac{2^{10}}{{10}^{10}}\)

\(\frac{1024}{{10}^{10}}\)

\(1024*10^{-10}\)

\(.0000001024\)

Tenth digit to the right of decimal is 4.

Ans: "C"

Hi fluke,

but, inasmuch as I know tenth digit to the right of the decimal point is a first number after the decimal point. Please correct if I am wrong!

(1/5)^10 is same as (2/10)^10. 10th digit after decimal in this is same as unit's digit in 2^10
2^10 = 2^5*2^5 - 32*32 whose units digit would be 4

You are talking about the 'tenths' digit which is right after the decimal point.
'The tenth digit to the right of the decimal' is the digit that appears after 9 digits to the right of the decimal point.
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(1/5)^10 is (0.2)^10

we know that 2^5=32 .we have (2^5) *(2^5) or 32*32=1024 since we have 10 zeros , our result is
0.0000001024

so 10th digit is 4
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Hi Karishma,

Thanks for reply. Such kind of things make GMAT very tricky. I appreciate your help.

I made a silly mistake...

Hi Karishma,
Thanks for making it clear . very confusing language.

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Yeah actually 1/5 = 0.2 = 2*10^-1

so 2^10 * 10^-10

2 has sequence 2,4,8,6.

Hence digit will be 4

Answer is C

Hope it helps

Cheers!
J

so, this is about Tenth vs Tenths! Tricky

1/5=0.2
Now the question becomes what is 10th digit of (0.2)¹⁰
for calculation remove decimal
2X2X2=8-a
2X2X2=8-b
(2X2X2)X2=8X2-c
8X8=64-a&b
64X8X2=1024-a&b&c
10th place after decimal
0.0000001024
Ans=4

hi

is it possible to find out the tenth digit of the expression without further simplification made to "2^10"....?
anybody out there ....?

thanks in advance ...

Analysis (30 seconds): The question is asking me to find the 10th digit of \(0.2^1^0\), the answer choices look familiar, possibly the pattern of final digits for powers of 2. I have absolutely no idea how to calculate the actual value of this so I'm going to go ahead and assume two things : 1) I can ignore the fact that the number is a decimal and focus on the 2, and 2) the 10th digit is actually the last digit (because I'm confident GMAC don't actually want me to compute the value of \(0.2^1^0\)). In order to solve this I'll quickly refresh my memory on the pattern of end digits for powers of 2 and then see what the 10th power would yield.

Strategy: Find the pattern, Count to 10

Find the pattern (30 seconds):
\(2^0 = 0\)
\(2^1 = 2\)
\(2^2 = 4\)
\(2^3 = 8\)
\(2^4 = 16\)
\(2^5 = 32\)
Looks like the pattern is: [2,4,8,6] with the exception of 0.

Count to 10 (10 seconds):
Using [2,4,8,6] as the pattern and starting from index 1 I can see that the 10th power will give me an end digit of 4.

Answer = C
Total Time: 1:10

1/5 = .2


Rule: when a Decimal is Raised to a Power, the No. of Decimal Places that will be in the Result will be =

(Number of Decimal Places in the BASE) * (Integer Exponent)


(1/5)^10 = (.2)^10

Base has = 1 Decimal Point

Exponent = 10th Power

1 * 10 = 10 Decimal Places are Required


(2)^10 = 1024 (memorize the Powers of 2 up to (2)^10)

move the Decimal Place 10 Places to the LEFT from 1024 ------

.000, 000, 1024

Digit in the 10th Place after the Decimal = 4

Hi,

asking for a specific digit to the right of the decimal usually allows us to use 10^x somehow, x being a positive integer. Taking a look at 5 and 10, we see the following pattern:

5 10 -> 1/2
5^2 10^2 -> 1/4
...

What this means is that 5^x=10^x*(1/2)^x

For our question, this means that:

5^10=10^10*(1/2)^10, now, multiplying by 1^-1 we get:

1/(5^10)=1/(10^10)*2^10=0.0000001024 -> (C)

In the given number :
\(\frac{1}{5^{10}}\)

The calculation part while converting fractions into decimal numbers can be reduced by converting the denominator into powers of 10.

In order to do this, we can rewrite the given expression as : \(\frac{2^{10}}{10^{10}}\)
This is equivalent to : \(\frac{1024}{10^{10}}\)
We can rewrite this in the decimal form as :
0.0000001024
Here the 10th digit to the right of the decimal place is 4.


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