Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

0. Review of Algebra

Radical Expressions

College Algebra

0. Review of Algebra

Radical Expressions

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0:38 minutes

Problem 63

Textbook Question

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

Verified step by step guidance

Identify the expression inside the parentheses: \( \frac{3a^{-5}b^2}{12a^3b^{-4}} \).

Recall the property of exponents: any non-zero number raised to the power of 0 is 1.

Apply this property to the entire expression: \( \left( \frac{3a^{-5}b^2}{12a^3b^{-4}} \right)^0 = 1 \).

Since the expression is raised to the power of 0, the simplified result is 1.

No further simplification is needed as the expression evaluates to 1.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.