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# Which of the following is equal to (2^k)(5^k − 1)?

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Which of the following is equal to (2^k)(5^k − 1)? [#permalink]

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11 Feb 2013, 08:50
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Which of the following is equal to 2^k*5^(k-1)?

A. 2*10^(k-1)
B. 5*10^(k-1)
C. 10^k
D. 2*10^k )
E. 10^(2k-1)
[Reveal] Spoiler: OA
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Re: Which of the following is equal to (2^k)(5^k − 1)? [#permalink]

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11 Feb 2013, 08:54
Expert's post
Which of the following is equal to 2^k*5^(k-1)?

A. 2*10^(k-1)
B. 5*10^(k-1)
C. 10^k
D. 2*10^k )
E. 10^(2k-1)

$$2^k*5^{k-1}=(2*2^{k-1})*5^{k-1}=2*10^{k-1}$$.

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Re: Which of the following is equal to (2^k)(5^k − 1)? [#permalink]

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20 Feb 2013, 18:16
Bunuel wrote:
Which of the following is equal to 2^k*5^(k-1)?

A. 2*10^(k-1)
B. 5*10^(k-1)
C. 10^k
D. 2*10^k )
E. 10^(2k-1)

$$2^k*5^{k-1}=(2*2^{k-1})*5^{k-1}=2*10^{k-1}$$.

I don't understand how you get 2*2^(K-1). I'm obviously missing something but can't figure it out. Can you please explain?
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Re: Which of the following is equal to (2^k)(5^k − 1)? [#permalink]

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20 Feb 2013, 21:15
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mp2469 wrote:
Bunuel wrote:
Which of the following is equal to 2^k*5^(k-1)?

A. 2*10^(k-1)
B. 5*10^(k-1)
C. 10^k
D. 2*10^k )
E. 10^(2k-1)

$$2^k*5^{k-1}=(2*2^{k-1})*5^{k-1}=2*10^{k-1}$$.

I don't understand how you get 2*2^(K-1). I'm obviously missing something but can't figure it out. Can you please explain?

$$2^3 = 2*2*2 = 2*2^2$$

Similarly, $$2^{10} = 2*2^9 = 2^2*2^8$$ etc

Hence $$2^k = 2*2^{k-1} = 2^2*2^{k-2} = 2^3*2^{k-3}$$ etc

Another Approach: Number Plugging.

Put k = 1 in $$2^k*5^{k-1}$$. You get $$2^1*5^0 = 2$$

When you put k = 1 in the options, only option (A) gives you 2.
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Math Expert Joined: 02 Sep 2009 Posts: 32657 Followers: 5659 Kudos [?]: 68754 [1] , given: 9818 Re: Which of the following is equal to (2^k)(5^k − 1)? [#permalink] ### Show Tags 21 Feb 2013, 03:20 1 This post received KUDOS Expert's post mp2469 wrote: Bunuel wrote: Which of the following is equal to 2^k*5^(k-1)? A. 2*10^(k-1) B. 5*10^(k-1) C. 10^k D. 2*10^k ) E. 10^(2k-1) $$2^k*5^{k-1}=(2*2^{k-1})*5^{k-1}=2*10^{k-1}$$. Answer: A. I don't understand how you get 2*2^(K-1). I'm obviously missing something but can't figure it out. Can you please explain? Operations involving the same exponents: Keep the exponent, multiply or divide the bases $$a^n*b^n=(ab)^n$$ Thus, $$2*2^{k-1}=2^{1+k-1}=2^k$$. For more check here: math-number-theory-88376.html Hope it helps. _________________ Current Student Joined: 21 Oct 2013 Posts: 194 Location: Germany GMAT 1: 660 Q45 V36 GPA: 3.51 Followers: 1 Kudos [?]: 26 [0], given: 19 Re: Which of the following is equal to (2^k)(5^k − 1)? [#permalink] ### Show Tags 17 Jan 2014, 05:06 Bunuel wrote: Operations involving the same exponents: Keep the exponent, multiply or divide the bases $$a^n*b^n=(ab)^n$$ Thus, $$2*2^{k-1}=2^{1+k-1}=2^k$$. For more check here: math-number-theory-88376.html Hope it helps. Hey Karishma, Hey Bunuel, Till now, I have encountered this kind of problem several times. Am I right to assume that these are the rules for simplifiying expontents like those in the questions: $$2^k=2*2^{k-1}$$ I can simplify from k to k-1. $$2^{k+1}=2*2^k$$. I can simplify from k+1 to k BUT I CAN'T simplify the first equation "backwards" meaning that if I see a exponent like $$5^{k-1}$$ I have to see directly that I have to get all other exponents to k-1?? I hope you get my question :D Thanks for your help Greetings! Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 6488 Location: Pune, India Followers: 1763 Kudos [?]: 10515 [0], given: 207 Re: Which of the following is equal to (2^k)(5^k − 1)? [#permalink] ### Show Tags 20 Jan 2014, 02:48 Expert's post unceldolan wrote: Bunuel wrote: Operations involving the same exponents: Keep the exponent, multiply or divide the bases $$a^n*b^n=(ab)^n$$ Thus, $$2*2^{k-1}=2^{1+k-1}=2^k$$. For more check here: math-number-theory-88376.html Hope it helps. Hey Karishma, Hey Bunuel, Till now, I have encountered this kind of problem several times. Am I right to assume that these are the rules for simplifiying expontents like those in the questions: $$2^k=2*2^{k-1}$$ I can simplify from k to k-1. $$2^{k+1}=2*2^k$$. I can simplify from k+1 to k BUT I CAN'T simplify the first equation "backwards" meaning that if I see a exponent like $$5^{k-1}$$ I have to see directly that I have to get all other exponents to k-1?? I hope you get my question :D Thanks for your help Greetings! What you need to do in any question depends on that particular question. You know that $$2^k=2*2^{k-1}$$ so you can easily get $$2^k$$ down to $$2^{k-1}$$. Also, $$2^{k-1} = 2^k/2$$. So whether you bring the terms down to (k-1) or (k) depends on the question. Here all options involve multiplication. Hence you will need to use $$2^k=2*2^{k-1}$$. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: Which of the following is equal to (2^k)(5^k − 1)? [#permalink]

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20 Jan 2014, 03:24
unceldolan wrote:
Bunuel wrote:

Operations involving the same exponents:
Keep the exponent, multiply or divide the bases
$$a^n*b^n=(ab)^n$$

Thus, $$2*2^{k-1}=2^{1+k-1}=2^k$$.

For more check here: math-number-theory-88376.html

Hope it helps.

Hey Karishma, Hey Bunuel,

Till now, I have encountered this kind of problem several times.
Am I right to assume that these are the rules for simplifiying expontents like those in the questions:

$$2^k=2*2^{k-1}$$ I can simplify from k to k-1.
$$2^{k+1}=2*2^k$$. I can simplify from k+1 to k

BUT I CAN'T simplify the first equation "backwards" meaning that if I see a exponent like $$5^{k-1}$$ I have to see directly that I have to get all other exponents to k-1??

I hope you get my question :D Thanks for your help

Greetings!

No, you could also change $$5^{k-1}$$ to $$\frac{5^{k}}{5}$$
It is a bit more complicated but may help to understand.

In this case, you would get
$$2^{k}*5^{k-1} = \frac{2^{k} * 5^{k}}{5} = \frac{10^{k}}{5} = \frac{10*10^{k-1}}{5} = 2*10^{k-1}$$
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Re: Which of the following is equal to (2^k)(5^k − 1)? [#permalink]

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Re: Which of the following is equal to (2^k)(5^k − 1)? [#permalink]

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13 Aug 2015, 00:36
4112019 wrote:
Which of the following is equal to 2^k*5^(k-1)?

A. 2*10^(k-1)
B. 5*10^(k-1)
C. 10^k
D. 2*10^k )
E. 10^(2k-1)

2^k*5^(k-1)=10^k*5^-1
option (A) is correct
2*10^(k-1) = 10^k*5^-1
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Re: Which of the following is equal to (2^k)(5^k − 1)? [#permalink]

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15 Aug 2015, 08:43
Bunuel wrote:
Which of the following is equal to 2^k*5^(k-1)?

A. 2*10^(k-1)
B. 5*10^(k-1)
C. 10^k
D. 2*10^k )
E. 10^(2k-1)

$$2^k*5^{k-1}=(2*2^{k-1})*5^{k-1}=2*10^{k-1}$$.

Is the following also correct ?

2^k x 5^(k-1) = 2^(k) x 5^(k) x 5^(-1)
= 10^(k)/5

?
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Re: Which of the following is equal to (2^k)(5^k − 1)? [#permalink]

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18 Aug 2015, 00:53
Expert's post
mike34170 wrote:
Bunuel wrote:
Which of the following is equal to 2^k*5^(k-1)?

A. 2*10^(k-1)
B. 5*10^(k-1)
C. 10^k
D. 2*10^k )
E. 10^(2k-1)

$$2^k*5^{k-1}=(2*2^{k-1})*5^{k-1}=2*10^{k-1}$$.

Is the following also correct ?

2^k x 5^(k-1) = 2^(k) x 5^(k) x 5^(-1)
= 10^(k)/5

?

Yes it is but it doesn't match any of the given options.
So you need to split the numerator as

$$10*10^{k - 1}/5 = 2*10^{k - 1}$$
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Re: Which of the following is equal to (2^k)(5^k − 1)? [#permalink]

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08 Mar 2016, 07:21
4112019 wrote:
Which of the following is equal to 2^k*5^(k-1)?

A. 2*10^(k-1)
B. 5*10^(k-1)
C. 10^k
D. 2*10^k )
E. 10^(2k-1)

Nice one

=> 2^k-1 * x 2 x 5^k-1 => 10^k-1 x 2 => option A
Re: Which of the following is equal to (2^k)(5^k − 1)?   [#permalink] 08 Mar 2016, 07:21
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