We are asked to find the value of $x$ in the equation $8^{2x+1} = \frac{1}{512}$.

1. Problem Description

We are asked to find the value of $x$ in the equation $8^{2x+1} = \frac{1}{512}$.

2. Solution Steps

We are given the equation $8^{2x+1} = \frac{1}{512}$.

We need to express both sides of the equation with the same base. Since $8 = 2^3$ and $512 = 2^9$, we can rewrite the equation as follows:

$(2^3)^{2x+1} = \frac{1}{2^9}$

Using the power of a power rule, we get

$2^{3(2x+1)} = 2^{-9}$

Since the bases are equal, we can equate the exponents:

$3(2x+1) = -9$

$6x+3 = -9$

Subtract 3 from both sides:

$6x = -9-3$

$6x = -12$

Divide both sides by 6:

$x = \frac{-12}{6}$

$x = -2$

3. Final Answer

$x = -2$