The expression x(x-y)5-y(x-y)5 can be rewritten as (x-y)a. What is the value of a?

We are given the expression:

\[
x(x - y)^5 - y(x - y)^5
\]

We need to rewrite it in the form \((x - y)^a\), and find the value of \(a\).

### Step-by-step process:

1. **Factor out** \((x - y)^5\) from both terms:
\[
x(x - y)^5 - y(x - y)^5 = (x - y)^5 (x - y)
\]

2. The remaining expression inside the parentheses is:
\[
(x - y)^5 (x - y)
\]

3. This is equivalent to:
\[
(x - y)^6
\]

Therefore, the expression can be rewritten as:

\[
(x - y)^6
\]

### Conclusion:
The value of \(a\) is \(6\).

Would you like a deeper explanation, or do you have any questions?

Here are 5 related questions for further exploration:
1. How do you factor expressions efficiently in algebra?
2. Can you explain the properties of exponents in algebraic expressions?
3. How can we apply the distributive property to complex algebraic expressions?
4. What are common mistakes when factoring and simplifying expressions?
5. How does this technique apply to polynomial expansions?

**Tip:** Always check for common factors before simplifying expressions; it makes the process faster and clearer.

The expression x(x-y)^5 - y(x-y)^5 can be rewritten as (x-y)^a. What is the value of a?

Explore how to factor and simplify the expression x(x-y)^5 - y(x-y)^5 by applying algebraic techniques and the distributive property. This problem is suitable for students in Grades 9-11 and focuses on key concepts like factoring and exponents.

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