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: minutes

Problem 14

Textbook Question

Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. √27

Verified step by step guidance

Recognize that \( \sqrt{27} \) can be rewritten using the property of square roots: \( \sqrt{a} = a^{1/2} \).

Express 27 as a product of its prime factors: \( 27 = 3^3 \).

Apply the property of exponents to rewrite \( \sqrt{27} \) as \( (3^3)^{1/2} \).

Use the power of a power property \((a^m)^n = a^{m \cdot n}\) to simplify \( (3^3)^{1/2} \) to \( 3^{3/2} \).

Recognize that \( 3^{3/2} \) can be further simplified to \( 3^{1} \cdot 3^{1/2} \), which is \( 3 \cdot \sqrt{3} \).

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Key Concepts

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

Product Rule

The product rule is a fundamental principle in algebra that states when multiplying two expressions, you can simplify the product by applying the rule to each factor. Specifically, if you have two functions, f(x) and g(x), the derivative of their product is given by f'(x)g(x) + f(x)g'(x). This rule is essential for simplifying expressions involving products of variables or functions.

Square Root Simplification

Simplifying square roots involves breaking down a square root into its prime factors to express it in a simpler form. For example, √27 can be simplified by recognizing that 27 = 9 × 3, leading to √27 = √(9 × 3) = √9 × √3 = 3√3. This concept is crucial for handling expressions that include square roots, especially when combined with other algebraic operations.

Nonnegative Real Numbers

Nonnegative real numbers are all real numbers that are either positive or zero. This concept is important in algebra as it restricts the domain of variables, ensuring that operations like square roots yield real results. When simplifying expressions, understanding that variables represent nonnegative values helps avoid complications that arise from negative inputs, particularly in square root calculations.