If \(2^x\cdot 2^y=8\) and \(9^x\cdot 3^y=81,\) then \((x,y)\) equals:

[M7MBA]

If \(2^x\cdot 2^y=8\) and \(9^x\cdot 3^y=81,\) then \((x,y)\) equals:

A. \((1,2)\)
B. \((2,1)\)
C. \((1,1)\)
D. \((2,2)\)
E. \((1,3)\)

[spoiler]OA=A[/spoiler]

Source: GMAT Prep

[Jay@ManhattanReview]

If \(2^x\cdot 2^y=8\) and \(9^x\cdot 3^y=81,\) then \((x,y)\) equals:

A. \((1,2)\)
B. \((2,1)\)
C. \((1,1)\)
D. \((2,2)\)
E. \((1,3)\)

[spoiler]OA=A[/spoiler]

Source: GMAT Prep

Given that \(2^x\cdot 2^y=8,\) we have \(2^{(x+y)}=2^3,\)

=> \(x+y=3\) ---(1)

similarly, \(9^x\cdot 3^y=81,\) we have
\(3^{(2x)}\cdot 3^y=3^4 => 3^{(2x+y)}=3^4,\)

=> \(2x+y=4\) ---(2)

From (1) and (2), we get x = 1 and y = 2.

Correct answer: A

Hope this helps!

-Jay
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[Scott@TargetTestPrep]

If \(2^x\cdot 2^y=8\) and \(9^x\cdot 3^y=81,\) then \((x,y)\) equals:

A. \((1,2)\)
B. \((2,1)\)
C. \((1,1)\)
D. \((2,2)\)
E. \((1,3)\)

[spoiler]OA=A[/spoiler]

Source: GMAT Prep

Solution:

Let’s simplify the given equations:

2^x * 2^y = 8

2^(x + y) = 2^3

x + y = 3 → Eq. 1

and

9^x * 3^y = 81

3^(2x + y) = 3^4

2x + y = 4 → Eq. 2

Subtracting Eq.1 form Eq. 2, we have:

x = 1

So y = 3 - x = 2 and (x, y) = (1, 2).

Answer: A