Understanding the Expression
To simplify (x^a/x^b)^(a^2 ab b^2), we start by breaking down the components of the expression.
Step 1: Simplify the Base
- The expression x^a/x^b can be simplified using the properties of exponents:
- x^a/x^b = x^(a-b)
Step 2: Rewrite the Expression
- Now, substitute the simplified base back into the expression:
- (x^(a-b))^(a^2 ab b^2)
Step 3: Apply Power of a Power Rule
- According to the power of a power property:
- (x^m)^n = x^(m*n)
- Therefore, we can rewrite our expression:
- x^((a-b) * (a^2 ab b^2))
Step 4: Final Expression
- The final simplified form of the original expression is:
- x^((a-b) * (a^2 ab b^2))
Key Takeaways
- The initial division of exponents simplifies to a single exponent.
- The power of a power rule allows further simplification.
- The final result combines both operations into a compact form.
Conclusion
This process demonstrates the efficient use of exponent rules to simplify complex expressions. Understanding these fundamental properties is essential for tackling various problems in mathematics, particularly in algebra and calculus.