We are given the equation $9^{z+5} = 3^z$ and we need to solve for $z$.

1. Problem Description

We are given the equation $9^{z+5} = 3^z$ and we need to solve for $z$.

2. Solution Steps

We can rewrite 9 as $3^2$. Substituting this into the given equation yields:

$(3^2)^{z+5} = 3^z$

Using the power of a power rule $(a^m)^n = a^{mn}$, we have

$3^{2(z+5)} = 3^z$

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

$2(z+5) = z$

Distribute the 2 on the left side:

$2z + 10 = z$

Subtract $2z$ from both sides:

$10 = z - 2z$

$10 = -z$

Multiply both sides by -1:

$z = -10$

3. Final Answer

The final answer is $z = -10$.