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

Question

What is the tenth digit to the right of the decimal point, in the decimal expansion of (1/5)^10
 (A) 0
 (B) 2
 (C) 4
 (D) 6
 (E) 8

fluke · Accepted Answer

(\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"

feruz77 · Answer

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!

beyondgmatscore · Answer

(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

KarishmaB · Answer

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.

LalaB · Answer

(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

Bunuel · Answer

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jlgdr · Answer

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

gps5441 · Answer

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

testcracker · Answer

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 ...

jameslewis · Answer

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:   with the exception of 0.

Count to 10 ( 10 seconds ): 
Using   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

Fdambro294 · Answer

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

hadimadi · Answer

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)

GmatPoint · Answer

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.
﻿

suhaanimanektala · Answer

Hi  jameslewis , can we be confident that in all such types of questions, they will be asking us for the units digit only and not actually compute the value? because this method seems far better and too good to be true maybe!

Krunaal · Answer

No, not necessarily. They could've asked the 7th digit or the first non-zero digit, and in that case your answer would be different (specifically 1). With practice, you will over time easily remember 1/5 = 0.2 and  2^{10}  = 1024. Then it's just  (0.2)^{10}  =  (2*10^{-1})^{10}  =  1024 * 10^{-10} , so 10th digit = 4. Possible to do it within 1:00 - 1:30 mins.

raunaknaikk · Answer

I used the cyclicity of 2 to get the answer-

(0.2)^10

Cyclicity of 2 is (2,4,8,6). And 10/4 leaves 2 as the remainder.

The second number in the cyclicity is 4 hence our answer is also 4. Hope this helps.

justdivyanshii · Answer

Hi Feruz,

the question kind of tricks you here. they are not talking about the "tenth, hundredth digit" they are talking about the "...8th,9th, 10th" digit from the right of the decimal.

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

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

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