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

[Vincen]

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

[Jay@ManhattanReview]

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]

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