1 over 2x to the power of 2 divided by 1 over 3x to the power of 2 plus 9x

We are given the expression:

\[
\frac{\frac{1}{2x^2}}{\frac{1}{3x^2} + 9x}
\]

### Step 1: Simplify the expression.
To simplify this, we first handle the division of the fractions.

\[
\frac{\frac{1}{2x^2}}{\frac{1}{3x^2} + 9x}
\]

The division of fractions is equivalent to multiplying by the reciprocal of the denominator. Thus, we have:

\[
= \frac{1}{2x^2} \times \frac{1}{\frac{1}{3x^2} + 9x}
\]

### Step 2: Simplify the complex fraction.
Now we simplify the expression in the denominator, which is:

\[
\frac{1}{3x^2} + 9x
\]

To combine these terms, we need a common denominator. The common denominator is \( 3x^2 \), so we can rewrite \( 9x \) as \( \frac{27x^3}{3x^2} \). Thus, we get:

\[
\frac{1}{3x^2} + 9x = \frac{1 + 27x^3}{3x^2}
\]

### Step 3: Substitute back.
Now substitute this into the original expression:

\[
\frac{1}{2x^2} \times \frac{1}{\frac{1 + 27x^3}{3x^2}} = \frac{1}{2x^2} \times \frac{3x^2}{1 + 27x^3}
\]

### Step 4: Simplify the result.
Now cancel out \( x^2 \) from the numerator and denominator:

\[
\frac{1}{2x^2} \times \frac{3x^2}{1 + 27x^3} = \frac{3}{2(1 + 27x^3)}
\]

### Final Answer:
The simplified expression is:

\[
\frac{3}{2(1 + 27x^3)}
\]

Let me know if you'd like further details or have any questions.

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Here are 5 related questions for practice:
1. How do you simplify complex fractions with variables?
2. What are the steps to find a common denominator when adding rational expressions?
3. How do you handle division of fractions?
4. What are the rules for canceling terms in algebraic expressions?
5. How does multiplying by the reciprocal help in simplifying expressions?

**Tip:** Always check for common denominators when adding or subtracting rational expressions!

Learn how to simplify a complex fraction involving variables: 1 over 2x^2 divided by 1 over 3x^2 plus 9x. This detailed step-by-step solution explains key algebraic concepts, including finding a common denominator, multiplying by the reciprocal, and simplifying algebraic fractions. Suitable for students in Grades 9-12.

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