Textbook Question
Simplify the algebraic expression. 5(2x - 3) + 7x
Ch. P - Fundamental Concepts of Algebra
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 1, Problem 18
Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers.
Verified step by step guidance1
Identify the two functions being multiplied: here, the expressions are \( \sqrt{10x} \) and \( \sqrt{8x} \).
Recall the product rule for square roots: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). Use this to combine the two square roots into one.
Multiply the expressions inside the square roots: \( 10x \times 8x = 80x^2 \). So, the expression becomes \( \sqrt{80x^2} \).
Simplify the square root by factoring out perfect squares. For example, \( 80 = 16 \times 5 \), and \( x^2 \) is a perfect square.
Rewrite the expression as \( \sqrt{16 \times 5 \times x^2} = \sqrt{16} \times \sqrt{5} \times \sqrt{x^2} \), then simplify each square root separately.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Product Rule for Radicals
The product rule for radicals states that the square root of a product equals the product of the square roots: √a * √b = √(a*b). This rule allows simplification by combining radicals under a single root when multiplying.
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Expanding Radicals
Properties of Exponents and Radicals
Radicals can be expressed as fractional exponents, such as √x = x^(1/2). Understanding this helps in manipulating and simplifying expressions involving roots and powers, especially when variables are involved.
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Domain Restrictions for Variables
When variables are under square roots, they must be nonnegative to keep the expression real. This restriction ensures the expression is defined within the real numbers and avoids complex results.
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Related Practice
Textbook Question
Evaluate each exponential expression in Exercises 1–22.
Textbook Question
In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. x2+8x+15
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Textbook Question
State the name of the property illustrated: (3 • 7) + (4 • 7) = (4 • 7) + (3 •7)
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Textbook Question
In Exercises 15–32, multiply or divide as indicated. (x2−4)/(x2−4x+4) ⋅ (2x−4)/(x+2)
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Textbook Question
Find each product. (x+7)(x+3)
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