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

Exponents

College Algebra

0. Review of Algebra

Exponents

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1:42 minutes

Problem 43

Textbook Question

Simplify each exponential expression in Exercises 23–64. (−3x^2 y^5)^2

Verified step by step guidance

Identify the expression to simplify: \((-3x^2 y^5)^2\).

Apply the power of a power property: \((a^m)^n = a^{m \cdot n}\).

Distribute the exponent 2 to each factor inside the parentheses: \((-3)^2, (x^2)^2, (y^5)^2\).

Calculate each part separately: \((-3)^2, x^{2 \cdot 2}, y^{5 \cdot 2}\).

Combine the results to form the simplified expression.

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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 are fundamental principles that govern the manipulation of expressions involving exponents. Key rules include the product of powers, power of a power, and power of a product. For example, when raising a power to another power, you multiply the exponents. Understanding these rules is essential for simplifying expressions like (−3x^2 y^5)^2.

Distributive Property

The distributive property is a key algebraic principle that allows you to multiply a single term by each term within a parenthesis. In the context of exponents, this means applying the exponent to each factor inside the parentheses. For instance, in (−3x^2 y^5)^2, you would distribute the exponent 2 to −3, x^2, and y^5, leading to a simplified expression.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the opposite positive exponent. While this concept is not directly applicable in the given expression, understanding it is crucial for broader exponential simplifications. For example, a term like x^(-n) can be rewritten as 1/(x^n), which is important when dealing with more complex expressions involving negative bases or exponents.