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We are given the equation:

\[
\left( \frac{1}{8b^4} \right) = 2^x a^y b^z
\]

and need to determine the values of \(x\), \(y\), and \(z\) that make this equation true. The problem also specifies that \(x = -3\), \(y = 0\), and we need to solve for \(z\).

### Step 1: Express the Left Side

The left side of the equation is:

\[
\frac{1}{8b^4} = 8^{-1} b^{-4}
\]

### Step 2: Express the Right Side

The right side of the equation is:

\[
2^x a^y b^z = 2^{-3} a^0 b^z
\]

Since \(a^0 = 1\), this simplifies to:

\[
2^{-3} b^z
\]

### Step 3: Compare the Powers of 2, \(a\), and \(b\)

Equating both sides, we need to match the powers of 2 and \(b\):

\[
8^{-1} b^{-4} = 2^{-3} b^z
\]

Note that \(8 = 2^3\), so:

\[
8^{-1} = (2^3)^{-1} = 2^{-3}
\]

Thus, we now have:

\[
2^{-3} b^{-4} = 2^{-3} b^z
\]

### Step 4: Solve for \(z\)

Since the powers of 2 are already equal, we equate the powers of \(b\):

\[
-4 = z
\]

Thus, \(z = -4\).

### Final Answer:

The values of \(x\), \(y\), and \(z\) are:
\[
x = -3, \quad y = 0, \quad z = -4
\]

Let me know if you need any further details or have questions! Here are some related questions for further practice:

1. What happens if the exponent of \(b\) on the right-hand side is positive?
2. How do you handle equations involving logarithmic expressions for unknown exponents?
3. Can you generalize this problem to a higher power of \(a\) in the equation?
4. How does the presence of a negative exponent affect the behavior of the function in a graph?
5. How do you solve similar equations with three different variables raised to unknown powers?

**Tip:** Whenever you are working with exponential equations, always focus on matching the bases and exponents separately to simplify the process!

Find the values of x, y, and z that make (1 / 8b^4) = 2^x a^y b^z true, where x = -3 and y = 0.

Learn how to solve the exponential equation (1 / 8b^4) = 2^x a^y b^z, where x = -3 and y = 0. This detailed step-by-step solution covers the core concepts of exponent rules and simplifying exponential expressions. Suitable for students in Grades 9-12.

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