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

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

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

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

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

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

\(.0000001024\)

Tenth digit to the right of decimal is 4.

Ans: "C"

(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

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!

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.
_________________

(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
_________________

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