Even Roots of Negative Exponents

1. Fundamental Concepts
Definition: A negative exponent represents a reciprocal, while an even root represents operations such as square roots or fourth roots.
Real Numbers: In the real number system, even roots are only defined for nonnegative bases.
Restriction: If the base is negative and the root index is even, the expression is undefined in real numbers.

2. Key Concepts
Exponent Rule: $(a^m)^{\tfrac{1}{n}} = a^{\tfrac{m}{n}}$ , valid for $a \geq 0$ when $n$ is even.
Negative Exponents: $a^{-m} = \tfrac{1}{a^m}$ .
Domain Restriction: For even roots, the base must be nonnegative to stay in the real number system.

3. Example
Undefined Case
Simplify: $(-9)^{\tfrac{1}{2}}$

This is undefined in real numbers because the base is negative and the root is even.

4. Problem-Solving Techniques
Check the base: If the base is negative and the root is even → undefined in real numbers.
Apply exponent rules carefully: Combine exponents before simplifying.
Domain awareness: Only nonnegative bases are valid for even roots in real numbers.