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

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


=> 2^k-1 * x 2 x 5^k-1 => 10^k-1 x 2 => option A
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\(2^k*5^{(k-1)}\)

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)...

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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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

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