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If 2^k + 2^(k-2) + 2^(k-3) = 2^(2k)*3^m, what is the value of k + m?​

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If 2^k + 2^(k-2) + 2^(k-3) = 2^(2k)*3^m, what is the value of k + m?​  [#permalink]

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New post Updated on: 15 Feb 2020, 20:22
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If \(2^k + 2^{k-2} - 2^{k-3} = 2^{2k}*3^m\), what is the value of k+m?​

A. -1​
B. 0​
C. 1​
D. 2​
E. 3 ​

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Originally posted by QuantMadeEasy on 15 Feb 2020, 02:33.
Last edited by QuantMadeEasy on 15 Feb 2020, 20:22, edited 2 times in total.
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Re: If 2^k + 2^(k-2) + 2^(k-3) = 2^(2k)*3^m, what is the value of k + m?​  [#permalink]

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New post 15 Feb 2020, 08:39
Top Contributor
kapil1 wrote:
If \(2^k + 2^{k-2} + 2^{k-3} = 2^{2k}*3^m\), what is the value of k+m?​

A. -1​
B. 0​
C. 1​
D. 2​
E. 3 ​


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Re: If 2^k + 2^(k-2) + 2^(k-3) = 2^(2k)*3^m, what is the value of k + m?​  [#permalink]

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New post 16 Apr 2020, 09:48
Top Contributor
kapil1 wrote:
If \(2^k + 2^{k-2} - 2^{k-3} = 2^{2k}*3^m\), what is the value of k+m?​

A. -1​
B. 0​
C. 1​
D. 2​
E. 3 ​


Given: \(2^k + 2^{k-2} - 2^{k-3} = 2^{2k}*3^m\)
Factor to get: \(2^{k-3}(2^3 + 2^1 - 1) = 2^{2k}*3^m\)
Evaluate: \(2^{k-3}(9) = 2^{2k}*3^m\)
Rewrite \(9\) as a power of \(3\) to get: \(2^{k-3}(3^2) = 2^{2k}*3^m\)

We can now conclude two things: \(2^{k-3}= 2^{2k}\) and \(3^2 = 3^m\)

If \(2^{k-3}= 2^{2k}\), then \(k-3= 2k\), which means \(k = -3\)
If \(3^2 = 3^m\), then \(m = 2\)

So, \(k+m = (-3) + 2 = -1\)

Answer: A

Cheers,
Brent
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Re: If 2^k + 2^(k-2) + 2^(k-3) = 2^(2k)*3^m, what is the value of k + m?​   [#permalink] 16 Apr 2020, 09:48

If 2^k + 2^(k-2) + 2^(k-3) = 2^(2k)*3^m, what is the value of k + m?​

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