How Do I solve this problem? (6z^4 +3 z^2 -9)(3z^2 -6)^-1  and what does the ^-1 mean/do the equation

Let's break down the expression step by step:

Given expression:  
\[
(6z^4 + 3z^2 - 9) \left( 3z^2 - 6 \right)^{-1}
\]

### Step 1: Understand the Meaning of \((3z^2 - 6)^{-1}\)
The term \((3z^2 - 6)^{-1}\) refers to the reciprocal (or inverse) of \(3z^2 - 6\). In other words, \((3z^2 - 6)^{-1}\) is the same as:
\[
\frac{1}{3z^2 - 6}
\]

### Step 2: Rewrite the Expression
Now that we know what \((3z^2 - 6)^{-1}\) means, we can rewrite the original expression as:
\[
\frac{6z^4 + 3z^2 - 9}{3z^2 - 6}
\]

### Step 3: Simplify the Denominator
Notice that the denominator, \(3z^2 - 6\), can be factored:
\[
3z^2 - 6 = 3(z^2 - 2)
\]

So the expression becomes:
\[
\frac{6z^4 + 3z^2 - 9}{3(z^2 - 2)}
\]

### Step 4: Factor the Numerator
Next, let's factor the numerator, \(6z^4 + 3z^2 - 9\). The first thing we can do is factor out a 3:
\[
6z^4 + 3z^2 - 9 = 3(2z^4 + z^2 - 3)
\]

Now the expression looks like this:
\[
\frac{3(2z^4 + z^2 - 3)}{3(z^2 - 2)}
\]

### Step 5: Cancel Common Factors
Since both the numerator and denominator have a factor of 3, we can cancel that:
\[
\frac{2z^4 + z^2 - 3}{z^2 - 2}
\]

### Step 6: Final Expression
The simplified expression is:
\[
\frac{2z^4 + z^2 - 3}{z^2 - 2}
\]

This is the simplified form of the given expression. It can't be simplified further unless you use specific values of \(z\).

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### Do you want further details or explanations on any part of this?

Here are 5 related questions you might find useful:
1. How do I solve equations with rational expressions?
2. What are some strategies for factoring polynomials?
3. How do you handle negative exponents in more complex expressions?
4. How can I simplify a rational expression when there are variables in both the numerator and denominator?
5. How do you solve equations with complex fractions?

**Tip:** Always check for common factors between the numerator and denominator before simplifying a rational expression.

How do I solve this problem? (6z^4 + 3z^2 -9)(3z^2 -6)^-1 and what does the ^-1 mean/do in the equation?

Learn how to simplify the rational expression (6z^4 + 3z^2 - 9)(3z^2 - 6)^-1 by understanding the meaning of negative exponents and using factoring techniques. This step-by-step solution covers algebraic concepts like polynomial factorization and simplification, suitable for students in Grades 9-12.

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