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What is the decimal equivalent of (2/5)^5 ?

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Bunuel
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What is the decimal equivalent of (2/5)^5 ? [#permalink] New post   07 Mar 2019, 01:49
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77% (01:17) correct 23% (01:17) wrong based on 308 sessions
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What is the decimal equivalent of \((\frac{2}{5})^5\) ?

A. 0.00016
B. 0.00032
C. 0.00256
D. 0.00512
E. 0.01024
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E
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Bunuel
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Re: What is the decimal equivalent of (2/5)^5 ? [#permalink] New post   07 Mar 2019, 01:50
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Bunuel wrote:
What is the decimal equivalent of \((\frac{2}{5})^5\) ?

A. 0.00016
B. 0.00032
C. 0.00256
D. 0.00512
E. 0.01024


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Re: What is the decimal equivalent of (2/5)^5 ? [#permalink] New post   07 Mar 2019, 01:54
Bunuel wrote:
What is the decimal equivalent of \((\frac{2}{5})^5\) ?

A. 0.00016
B. 0.00032
C. 0.00256
D. 0.00512
E. 0.01024



Given,

\((\frac{2}{5})^5\)

= \((0.4)^5\)

= (0.4) (0.4) (0.4) (0.4) (0.4) (0.4)

=0.01024.

E is the correct answer.
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Re: What is the decimal equivalent of (2/5)^5 ? [#permalink] New post   07 Mar 2019, 01:57
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(2/5)^5 can be written as (4/10)^5

Which is:

2^10
100000

Answer is 0.01024 (E)
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What is the decimal equivalent of (2/5)^5 ? [#permalink] New post   07 Mar 2019, 02:00
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On multiplying numerator and denominator by 2^5:
(2/5)^5 * (2/2)^5
We get numerator as = 2^10
denominator as = (2*5)^5

2^10 = 1024
(2*5)^5 = 10 ^5

Ans: 0.01024

Tip: Always try to convert denominator as multiple of 10.
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Re: What is the decimal equivalent of (2/5)^5 ? [#permalink] New post   07 Mar 2019, 02:02
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=(2/5)^{5}
=(4/10)^{5}
=4^{5}/10^{5}
=1024/10^{5}
=0.01024
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Re: What is the decimal equivalent of (2/5)^5 ? [#permalink] New post   09 Mar 2019, 14:40
Some good explanation above.
I will throw another idea out there-

We know we can write the term as \((\frac{4}{10}) ^5\)

Which means \((\frac{4*4*4*4*4}{100000})\)

Cyclicity of 4 is 2. Meaning- If you keep multiplying 4 by 4, the units digits keep changing between 6 and 4. Since the power is 5 it's cyclicity is 2m+1 which tells us the unit's digit will be 4.
Or you can just multiple 4 to itself 5 times in your head (keeping track of just the unit's digit- 4*4 = 16 ; 6*4 = 24; 4*4 = 16 ; 6*4 = 24 ) Unit's digit is 4.

Option E is the only option with unit's digit 4.
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Re: What is the decimal equivalent of (2/5)^5 ? [#permalink] New post   14 Feb 2022, 19:13
Here is the solution-
First we divide 2/5, we get 0.4

We will get (0.4)^5.

Now, we know that if we multiply 0.4 * 0.4 we get unit digit 6 that means 0.4^4 the unit digit will be 6.

Now if we multiply 6 with the remaining 0.4, we get 4 as unit digit. This means that the answer should contain the unit digit 4.

So Option E would be our answer
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Re: What is the decimal equivalent of (2/5)^5 ? [#permalink] New post   23 Apr 2023, 00:06
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30 second approach:

The units digits of powers of 4 follow a two-number pattern: 4 and 6.

All positive odd powers of 4 end in 4, and all positive even powers of 4 end in 6.

In this question, 0.4 is raised to an odd power, therefore the units digit of the result must end in 4.

Only E fits the bill.
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Re: What is the decimal equivalent of (2/5)^5 ? [#permalink]
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