If 3^(2n) = (1/9)^(n+2), what is the value of n?

Question

If  3^{(2n)} = (\frac{1}{9})^{(n+2)} , what is the value of n?

A. -2
B. -1
C. 0
D. 1
E. 2

BrentGMATPrepNow · Accepted Answer

For these kinds of questions, we typically must rewrite the terms so that they have the SAME BASES. 
One option is to rewrite the right hand side with a power of 3.

Given: 3^(2n) =  (1/9) ^(n + 2)
Rewrite  1/9  to get: 3^(2n) =    ^(n + 2)
Apply Power of a Power law to get: 3^(2n) = 3^(-2n - 4)
We can now conclude that 2n = -2n - 4
Solve, to get: n = -1
Answer: B

Cheers,
Brent

Kurtosis · Answer

3^(2n) = 3^(-2n - 4)
Equate the powers: 2n = -2n - 4
4n = -4
n = -1

Answer: B

deabas · Answer

1/9 is the reciprocal of 3 so the exponent of 3 should negative -> answer choices C, D, and E are eliminated.

By replacements:
Left-hand side: 3^-2 = 1/3^2 = 1/9
Right-hand side: (1/9)^2+(-1) = (1/9)^1 = 1/9

The correct answer choice is B

buan15 · Answer

1/9= 3^-2 So, 3^(2n) = 3^(-2n - 4) Equating the powers: 2n = -2n - 4 4n = -4 n = -1 B is the answer

ScottTargetTestPrep · Answer

Using the negative exponent rule, we have:

3^2n = 9^(-n - 2)

3^2n = 3^(-2n - 4)

With a common base of 3 for both expressions, we can equate the exponents:

2n = -2n - 4

4n = -4

n = -1

Answer: B

GMATWhizTeam · Answer

Since the answer options represent the value of n, back-solving can be a great strategy to solve this question.

But before starting with the options, let us simplify the equation given to ensure that we have the same bases on both sides.

3^{2n}  can be written as  (3^2)^n  i.e.  9^n 
. 
 (\frac{1}{9})^{n+2}  can be written as  \frac{1 }{ 9^{n+2}}  since 1 raised to any power is always 1.

Therefore, the equation can be rewritten as,  9^n  = \frac{ 1 }{ 9^{n+2}}

Let’s start with answer option C i.e. n = 0. If n = 0, LHS =  9^0  = 1 and RHS =  \frac{1 }{ 9^3} .

Clearly, both are not equal since the LHS is an integer and the RHS is a fraction. Answer option C can be eliminated. Further, it helps us understand that we cannot use any positive value for n since it will create a similar situation. 
This helps us eliminate answer options D and E. We are left with answer options A and B.

If n = -1, LHS =  9^{-1}  =  \frac{1}{9}  and RHS =  \frac{1 }{ 9^{1}}  =  \frac{1}{9} . LHS = RHS, so the equation is satisfied.

Answer option A can be eliminated since there can only be one right answer. 
 The correct answer option is B.

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If 3^(2n) = (1/9)^(n+2), what is the value of n?

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If 3^(2n) = (1/9)^(n+2), what is the value of n?

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