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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]  11 Feb 2013, 07:50
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68% (02:30) correct 32% (01:40) wrong based on 81 sessions
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
Veritas Prep GMAT Instructor
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Location: Pune, India
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Kudos [?]: 6795 [2] , given: 177

Re: Which of the following is equal to (2^k)(5^k − 1)? [#permalink]  20 Feb 2013, 20:15
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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)

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

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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Karishma
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Math Expert
Joined: 02 Sep 2009
Posts: 27228
Followers: 4231

Kudos [?]: 41092 [1] , given: 5666

Re: Which of the following is equal to (2^k)(5^k − 1)? [#permalink]  21 Feb 2013, 02:20
1
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)

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

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.
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Math Expert
Joined: 02 Sep 2009
Posts: 27228
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Kudos [?]: 41092 [0], given: 5666

Re: Which of the following is equal to (2^k)(5^k − 1)? [#permalink]  11 Feb 2013, 07:54
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)

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

_________________
Intern
Joined: 12 Dec 2012
Posts: 3
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Kudos [?]: 0 [0], given: 3

Re: Which of the following is equal to (2^k)(5^k − 1)? [#permalink]  20 Feb 2013, 17:16
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?
Manager
Joined: 21 Oct 2013
Posts: 194
Location: Germany
GMAT 1: 660 Q45 V36
GPA: 3.51
Followers: 0

Kudos [?]: 18 [0], given: 19

Re: Which of the following is equal to (2^k)(5^k − 1)? [#permalink]  17 Jan 2014, 04:06
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: 5459
Location: Pune, India
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Kudos [?]: 6795 [0], given: 177

Re: Which of the following is equal to (2^k)(5^k − 1)? [#permalink]  20 Jan 2014, 01:48
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

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Intern
Status: Student
Joined: 06 Oct 2013
Posts: 29
Location: Germany
Concentration: Operations, General Management
GMAT 1: 670 Q49 V35
GPA: 2.4
WE: Other (Consulting)
Followers: 1

Kudos [?]: 21 [0], given: 17

Re: Which of the following is equal to (2^k)(5^k − 1)? [#permalink]  20 Jan 2014, 02:24
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] 20 Jan 2014, 02:24
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