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

Question

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)}

Bunuel · Accepted Answer

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.

KarishmaB · Answer

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.

mp2469 · Answer

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?

Bunuel · Answer

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.

unceldolan · Answer

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!

KarishmaB · Answer

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} .

christianbze · Answer

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}

mathivanan · Answer

2^k*5^(k-1)=10^k*5^-1
option (A) is correct
2*10^(k-1) = 10^k*5^-1

mike34170 · Answer

Is the following also correct ?

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

?

KarishmaB · Answer

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}

stonecold · Answer

Nice one

=> 2^k-1 * x 2 x 5^k-1 => 10^k-1 x 2 => option A

sashiim20 · Answer

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}

Answer (A)...

KanishkM · Answer

Question is

2^k   5^{k-1}

2^k   5^k ] / 5

A) 2 * 10^k * 1/10

2^k   5^k ] / 5

ashu0318 · Answer

2^k*5^(k-1)
2^k*5^k/5
(2*5)^k/5
10^k/5

multiple by 2 on both up and down
2*10^k/10
2*10^(k-1)

Ans: A

v12345 · Answer

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

Hence, OA is (A).

MclLaurent · Answer

Hi  Bunuel ,

Please add gmat focus tag for this question, just test from Focus trial testing.

Bunuel · Answer

_____________________
Done. Thank you!

A_Nishith · Answer

2k∗5k−1=(2∗2k−1)∗5k−1=2∗10k−1

Answer: A.

Chris01 · Answer

Assume k=2

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

2^2*5^{(2-1)}   =20

Plug in k=2 into the answer choices and see which one yields 20.

A.

2*10^{(2-1)}  = 20

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)?

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