Question

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?(1 point)
Responses

x=1/4
x= -5+sqrt73/8
x=-2
x=-4

Answer 1

To solve the equation \( 3^{-2x+1} \cdot 3^{-2x-3} = 3^{-3} \), we can first simplify the left side using the properties of exponents:

\[
3^{-2x+1} \cdot 3^{-2x-3} = 3^{(-2x+1) + (-2x-3)} = 3^{-4x - 2}
\]

Thus, the equation becomes:

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

Since the bases are the same, 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:

\[
\boxed{\frac{1}{4}}
\]
Thus, Esmeralda should select \( \frac{1}{4} \) as her answer.