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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1:35 minutes

Problem 41

Textbook Question

Use the product rule to simplify the expressions in Exercises 41 - 44. In exercises 43 - 44, assume that variables represent nonnegative real numbers. √300

Verified step by step guidance

Identify the expression to simplify: \( \sqrt{300} \).

Express 300 as a product of its prime factors: \( 300 = 2^2 \times 3 \times 5^2 \).

Apply the product rule for square roots: \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \).

Simplify \( \sqrt{300} \) using the prime factorization: \( \sqrt{2^2 \times 3 \times 5^2} = \sqrt{2^2} \times \sqrt{3} \times \sqrt{5^2} \).

Simplify each square root: \( \sqrt{2^2} = 2 \) and \( \sqrt{5^2} = 5 \), then multiply the results with \( \sqrt{3} \).

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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 calculus used to differentiate products of functions. It states that if you have two functions, u(x) and v(x), the derivative of their product is given by u'v + uv'. This rule is essential for simplifying expressions involving products, especially when dealing with variables and constants.

Square Root Simplification

Simplifying square roots involves breaking down a square root into its prime factors to make calculations easier. For example, √300 can be simplified by recognizing that 300 = 100 × 3, leading to √300 = √(100 × 3) = 10√3. This concept is crucial for handling expressions involving square roots in algebra.

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. Understanding this helps in correctly interpreting and simplifying expressions that involve variables representing nonnegative values.