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Ch. P - Fundamental Concepts of Algebra
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. P - Fundamental Concepts of AlgebraProblem 22
Chapter 1, Problem 22

Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. 6x⋅3x2\(\sqrt{6x}\[\cdot\]\sqrt{3x^2}\)

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Identify the two functions being multiplied: here, the first function is \(\sqrt{6x}\) and the second function is \(\sqrt{3x^2}\).
Recall that the product rule for derivatives states: \(\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)\), but since the problem asks to simplify the product, focus on simplifying the expression \(\sqrt{6x} \cdot \sqrt{3x^2}\) first.
Use the property of square roots that \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\) to combine the two square roots into one: \(\sqrt{6x} \cdot \sqrt{3x^2} = \sqrt{(6x)(3x^2)}\).
Multiply inside the square root: \((6x)(3x^2) = 18x^3\), so the expression becomes \(\sqrt{18x^3}\).
Simplify \(\sqrt{18x^3}\) by factoring inside the root to extract perfect squares: write \$18x^3$ as \(9 \cdot 2 \cdot x^2 \cdot x\), then use \(\sqrt{a^2} = a\) to simplify.

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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, i.e., √a * √b = √(a*b). This rule allows simplification by combining radicals under a single square root when multiplying.
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Properties of Exponents

When multiplying expressions with the same base, add their exponents: x^m * x^n = x^(m+n). This property helps simplify terms like x and x^2 inside or outside radicals.
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Simplifying Radicals with Variables

To simplify radicals containing variables, express variables with exponents inside the root and apply exponent rules. For nonnegative variables, √(x^2) simplifies to x, ensuring the expression remains valid.
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