$$ \frac{\left(2x-4\right)^5}{\left(2-x\right)^5} $$ simplify

To simplify the expression 

\[
\frac{\left(2x-4\right)^5}{\left(2-x\right)^5},
\]

we can start by observing the relationship between the terms in the numerator and denominator.

Notice that $2x - 4$ can be factored as:

\[
2x - 4 = 2(x - 2).
\]

Thus, the expression becomes:

\[
\frac{(2(x - 2))^5}{\left(2 - x\right)^5}.
\]

We can rewrite this as:

\[
\frac{2^5(x - 2)^5}{(2 - x)^5}.
\]

Next, observe that $2 - x$ can be rewritten as $-(x - 2)$. Therefore:

\[
\frac{2^5(x - 2)^5}{(-(x - 2))^5}.
\]

Since raising $-1$ to an odd power (like 5) gives $-1$, this simplifies to:

\[
\frac{2^5(x - 2)^5}{-1 \cdot (x - 2)^5}.
\]

Finally, the $(x - 2)^5$ terms cancel out:

\[
\frac{2^5}{-1} = -32.
\]

So, the simplified expression is:

\[
\boxed{-32}.
\]

Would you like more details or have any other questions?

Here are some related questions you might find interesting:
1. How do you factor expressions like $2x - 4$ in general?
2. What is the general rule for simplifying fractions with similar terms in the numerator and denominator?
3. How does the property of negative exponents apply in simplifications?
4. What happens if the exponent is even in a similar problem?
5. How do you handle more complex expressions with multiple terms in both the numerator and denominator?

**Tip:** When simplifying expressions, always look for common factors or ways to factorize terms before canceling or applying other algebraic operations.

Learn how to simplify the expression (2x-4)^5 / (2-x)^5 with a detailed step-by-step solution. This problem involves algebraic manipulations, including factoring and simplifying fractions with negative exponents. Suitable for students in Grades 10-12.

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