Simplify each exponential expression in Exercises 23–64. (3a^−5 b^2/12a^3 b^−4)^0

Verified step by step guidance

1. Identify the expression inside the parentheses: \( \frac{3a^{-5}b^2}{12a^3b^{-4}} \).
2. Recall the property of exponents: any non-zero number raised to the power of 0 is 1.
3. Apply this property to the entire expression: \( \left( \frac{3a^{-5}b^2}{12a^3b^{-4}} \right)^0 = 1 \).
4. Since the expression is raised to the power of 0, the simplified result is 1.
5. No further simplification is needed as the expression evaluates to 1.

Key Concepts

Exponential Rules

Exponential rules govern how to manipulate expressions involving exponents. Key rules include the product of powers (a^m * a^n = a^(m+n)), the power of a power ( (a^m)^n = a^(m*n)), and the zero exponent rule (a^0 = 1 for any a ≠ 0). Understanding these rules is essential for simplifying expressions with exponents.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form. This process often includes combining like terms, applying the distributive property, and using exponent rules. In the context of the given expression, recognizing that any non-zero base raised to the power of zero equals one is crucial for simplification.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the opposite positive exponent (a^−n = 1/a^n). This concept is vital when simplifying expressions that contain negative exponents, as it allows for rewriting the expression in a more manageable form. Understanding how to handle negative exponents is key to solving the given problem.