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Intern  Joined: 10 Jan 2013
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Which of the following is equal to (2^k)(5^k − 1)?  [#permalink]

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Question Stats: 70% (01:32) correct 30% (01:57) wrong based on 202 sessions

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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)
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Posts: 58335
Re: Which of the following is equal to (2^k)(5^k − 1)?  [#permalink]

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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)

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

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

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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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Joined: 16 Oct 2010
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Re: Which of the following is equal to (2^k)(5^k − 1)?  [#permalink]

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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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Math Expert V
Joined: 02 Sep 2009
Posts: 58335
Re: Which of the following is equal to (2^k)(5^k − 1)?  [#permalink]

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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?

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

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

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

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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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Manager  Joined: 10 Jun 2015
Posts: 111
Re: Which of the following is equal to (2^k)(5^k − 1)?  [#permalink]

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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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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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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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GRE 1: Q169 V154 Re: Which of the following is equal to (2^k)(5^k − 1)?  [#permalink]

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

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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)

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

Simplifying the expression we get;

$$2^k*\frac{5^k}{5}$$

$$\frac{2^k*5^k}{5}$$

$$\frac{(2*5)^k}{5} = \frac{10^k}{5}$$

Check the options;

(A) $$2*10^{(k-1)} = 2*\frac{10^k}{10} = \frac{10^k}{5}$$

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

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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)

Question is

$$2^k$$ $$5^{k-1}$$

[$$2^k$$ $$5^k$$] / 5

A) 2 * 10^k * 1/10

[$$2^k$$ $$5^k$$] / 5
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Many of life's failures happen with people who do not realize how close they were to success when they gave up. Re: Which of the following is equal to (2^k)(5^k − 1)?   [#permalink] 10 Feb 2019, 10:38
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