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Let's break down the given expression step by step.

The expression is:

\[
\frac{\frac{3}{b^2} - \frac{1}{b^4}}{\frac{1}{b^3}}
\]

### Step 1: Simplify the numerator
The numerator is:

\[
\frac{3}{b^2} - \frac{1}{b^4}
\]

To subtract these fractions, we need a common denominator. The common denominator between \(b^2\) and \(b^4\) is \(b^4\). Rewrite each term:

\[
\frac{3}{b^2} = \frac{3b^2}{b^4}, \quad \frac{1}{b^4} = \frac{1}{b^4}
\]

Now subtract the two fractions:

\[
\frac{3b^2 - 1}{b^4}
\]

### Step 2: Simplify the whole expression
Now, the entire expression becomes:

\[
\frac{\frac{3b^2 - 1}{b^4}}{\frac{1}{b^3}}
\]

We can simplify this by multiplying the numerator by the reciprocal of the denominator:

\[
= \left( \frac{3b^2 - 1}{b^4} \right) \times \left( \frac{b^3}{1} \right)
\]

This simplifies to:

\[
\frac{b^3(3b^2 - 1)}{b^4} = \frac{3b^5 - b^3}{b^4}
\]

### Step 3: Simplify the final expression
Now simplify each term by canceling the powers of \(b\):

\[
= \frac{3b^5}{b^4} - \frac{b^3}{b^4}
\]

Simplifying both terms:

\[
= 3b - \frac{1}{b}
\]

Thus, the simplified expression is:

\[
3b - \frac{1}{b}
\]

Would you like further details on any of these steps or more explanation? 

Here are 5 questions for deeper understanding:
1. How do you find a common denominator when subtracting fractions?
2. Why do we multiply by the reciprocal of the denominator when simplifying complex fractions?
3. How are exponents subtracted when dividing terms with the same base?
4. What happens to the expression if we start with different exponents?
5. Can you generalize the method used here for other variables or exponents?

**Tip:** When simplifying expressions involving fractions, always look for a common denominator and work through one step at a time to avoid mistakes.

Simplify the expression: (3/b^2 - 1/b^4) / (1/b^3), and write the answer using only positive exponents.

Learn how to simplify the expression (3/b^2 - 1/b^4) / (1/b^3) using exponent rules and algebraic fraction manipulation. This problem covers important algebra concepts like finding common denominators, applying exponent laws, and simplifying complex rational expressions. Suitable for students in Grades 9-11.

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