2.a-4.b4     (3.a-2.b3)2                                                                                                                                                                           

It seems like you have a complex algebraic expression. Let me break it down and simplify it for you step by step.

The expression is:

\[
\frac{2a - 4b^4}{(3a - 2b^3)^2}
\]

### Step 1: Simplify numerator and denominator
- Numerator: \( 2a - 4b^4 \) (This cannot be simplified further at this point.)
- Denominator: \( (3a - 2b^3)^2 \) (We need to square the binomial.)

### Step 2: Expand the denominator
Using the identity \((x - y)^2 = x^2 - 2xy + y^2\), we expand:

\[
(3a - 2b^3)^2 = (3a)^2 - 2(3a)(2b^3) + (2b^3)^2
\]

\[
= 9a^2 - 12ab^3 + 4b^6
\]

### Step 3: Final expression
Now, we can rewrite the original expression as:

\[
\frac{2a - 4b^4}{9a^2 - 12ab^3 + 4b^6}
\]

This is the simplified form of your given algebraic expression.

Would you like a deeper explanation or a different method of solving this?

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### Related Questions:
1. How do you expand binomials using the square formula?
2. What are common ways to factor polynomials?
3. How can we simplify expressions with exponents?
4. What are the steps to apply the quotient rule in algebraic simplification?
5. How do you apply the distributive property in polynomial multiplication?

#### Tip:
When dealing with complex algebraic expressions, breaking them into smaller parts often makes simplification easier.

Learn how to simplify the rational expression 2a - 4b^4 / (3a - 2b^3)^2 with a detailed step-by-step solution. This problem covers key algebraic concepts such as rational expressions, binomial expansion, and exponent rules. Suitable for students in Grades 9-12.

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