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

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) = [3^(-2)]^(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

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

Answer: B

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

1/9= 3^-2

So, 3^(2n) = 3^(-2n - 4)
Equating the powers: 2n = -2n - 4
4n = -4
n = -1

B is the answer
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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
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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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