If $8^x\cdot 9^{2y}=81\cdot 2^{12y},$ what is the value of $x?$

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

by Vincen » Sun May 03, 2020 12:15 pm

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If $8^x\cdot 9^{2y}=81\cdot 2^{12y},$ what is the value of $x?$

(A) 2
(B) 4
(C) 8
(D) 12
(E) 16

[spoiler]OA=B[/spoiler]

Source: Veritas Prep

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Jay@ManhattanReview: GMAT Instructor; Posts: 3008; Joined: Mon Aug 22, 2016 6:19 amFollowed by: 34 members; Location: Grand Central / New York; Thanked: 470 times

Re: If $8^x\cdot 9^{2y}=81\cdot 2^{12y},$ what is the value of $x?$

by Jay@ManhattanReview » Sun May 03, 2020 10:43 pm

Vincen wrote: ↑
Sun May 03, 2020 12:15 pm
If $8^x\cdot 9^{2y}=81\cdot 2^{12y},$ what is the value of $x?$

(A) 2
(B) 4
(C) 8
(D) 12
(E) 16

[spoiler]OA=B[/spoiler]

Source: Veritas Prep

For $8^x\cdot 9^{2y}=81\cdot 2^{12y},$ let's factorize 8, 9, 81, and 12 into prime factors.

We have $8^x\cdot 9^{2y}=81\cdot 2^{12y}$

$2^{3x}\cdot 3^{4y}=3^4\cdot 2^{12y}$

=> 3x = 12y => x = 4y; and 4y = 4 => y = 1. Thus, x = 4y = 4*1 = 4.

The correct answer: B

Hope this helps!

-Jay
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Scott@TargetTestPrep: GMAT Instructor; Posts: 8156; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

Re: If $8^x\cdot 9^{2y}=81\cdot 2^{12y},$ what is the value of $x?$

by Scott@TargetTestPrep » Sat May 09, 2020 5:45 am

Vincen wrote: ↑
Sun May 03, 2020 12:15 pm
If $8^x\cdot 9^{2y}=81\cdot 2^{12y},$ what is the value of $x?$

(A) 2
(B) 4
(C) 8
(D) 12
(E) 16

[spoiler]OA=B[/spoiler]

Source: Veritas Prep

Simplifying, we have:

2^3x * 3^4y = 3^4 * 2^12y

Equating the respective exponents of 2 and 3, we have:

3x = 12 y

x = 4y

and

4y = 4

y = 1

Since x = 4y, we have x = 4 * 1 = 4.

Answer: B

Scott Woodbury-Stewart
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