Esmeralda wants to solve for x in the equation 3^−2x+1*3^−2x−3=3^−3 . Which of the following answers should she select?

Answer 1

To solve the equation \( 3^{-2x+1} \cdot 3^{-2x-3} = 3^{-3} \), we start by using the properties of exponents.

First, we can combine the exponents on the left-hand side:

\[ 3^{-2x+1} \cdot 3^{-2x-3} = 3^{(-2x+1) + (-2x-3)} \]

Now we simplify the exponent:

\[ (-2x+1) + (-2x-3) = -2x - 2x + 1 - 3 = -4x - 2 \]

Thus, the equation becomes:

\[ 3^{-4x-2} = 3^{-3} \]

Since we have the same base, we can set the exponents equal to each other:

\[ -4x - 2 = -3 \]

Now, solve for \( x \):

1. Add 2 to both sides:

\[ -4x = -3 + 2 \]

\[ -4x = -1 \]

2. Divide by -4:

\[ x = \frac{-1}{-4} = \frac{1}{4} \]

So, the solution for \( x \) is:

\[ x = \frac{1}{4} \]

Esmeralda should select \( \frac{1}{4} \) as the final answer.