\[ If\ \ 3^k+3^k=(3^9)^{3^9}-3^k,\ \ \ then\ \ \ \ k=? \]

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VJesus12: Legendary Member; Posts: 2276; Joined: Sat Oct 14, 2017 6:10 am; Followed by:3 members

\[ If\ \ 3^k+3^k=(3^9)^{3^9}-3^k,\ \ \ then\ \ \ \ k=? \]

by VJesus12 » Thu Jun 14, 2018 1:05 am

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$$If\ \ 3^k+3^k=(3^9)^{3^9}-3^k,\ \ \ then\ \ \ \ k=?$$ (A) 11/3
(B) 11/2
(C) 242
(D) 3^10
(E) 3^11 - 1

The OA is the option E.

How can I rewrite the given expression to get the correct answer? I could solve the powers.

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Source: Beat The GMAT — Problem Solving | Thu Jun 14, 2018 1:05 am

Vincen: Legendary Member; Posts: 2898; Joined: Thu Sep 07, 2017 2:49 pm; Thanked: 6 times; Followed by:5 members

by Vincen » Thu Jun 14, 2018 2:01 am

VJesus12 wrote:$$If\ \ 3^k+3^k=(3^9)^{3^9}-3^k,\ \ \ then\ \ \ \ k=?$$ (A) 11/3
(B) 11/2
(C) 242
(D) 3^10
(E) 3^11 - 1

The OA is the option E.

How can I rewrite the given expression to get the correct answer? I could solve the powers.

Hello Vjesus12.

You can rewrite the powers as follows: $$3^k+3^k=(3^9)^{3^9}-3^k\ \ \ \Rightarrow\ \ \ \ 2\cdot3^k=\left(3\right)^{9\cdot3^9}-3^k$$ $$\Rightarrow\ \ \ \ 2\cdot3^k+3^k=\left(3\right)^{3^2\cdot3^9}$$ $$\Rightarrow\ \ \ \ 3\cdot3^k=\left(3\right)^{3^{11}}$$ $$\Rightarrow\ \ \ \ 3^{k+1}=\left(3\right)^{3^{11}}$$ $$\Rightarrow\ \ k+1=3^{11}$$ $$\Rightarrow\ \ k=3^{11}-1.$$ That way we get that the correct answer is the option E .

I hope it helps you.

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DrMaths: Senior | Next Rank: 100 Posts; Posts: 82; Joined: Mon Jan 15, 2018 2:01 am

by DrMaths » Thu Jun 14, 2018 3:41 am

3^k+3^k=(3^9)^3^9âˆ’3k
3^k+3^k+3^k=(3^3^2)^3^9
3(3^k) = (3)^3^(2+9)
3(3^k) = 3^3^(11)
(3^k) = 3^3^(11)-1
k = 3^(11)-1

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Jeff@TargetTestPrep: GMAT Instructor; Posts: 1462; Joined: Thu Apr 09, 2015 9:34 am; Location: New York, NY; Thanked: 39 times; Followed by:22 members