Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. √14 • √3pqr

Verified step by step guidance

1. Recall the product rule for radicals, which states that \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \) for positive real numbers \(a\) and \(b\).
2. Apply the product rule to the given expression: \( \sqrt{14} \cdot \sqrt{3pqr} = \sqrt{14 \cdot 3pqr} \).
3. Multiply the numbers inside the radical: \( 14 \cdot 3 = 42 \), so the expression becomes \( \sqrt{42pqr} \).
4. Since all variables represent positive real numbers, you can leave the expression under one radical without needing to simplify further.
5. Write the final simplified expression as \( \sqrt{42pqr} \).

Key Concepts

Properties of Radicals

Radicals represent roots, commonly square roots, and follow specific properties such as the product rule: √a * √b = √(a*b). This allows multiplication of radicals by combining their radicands under a single root, simplifying expressions efficiently.

Multiplication of Variables under Radicals

When variables are under radicals, their multiplication follows the same product rule. For positive variables, √(p) * √(q) = √(p*q), enabling the combination of variable expressions inside one radical, which simplifies the expression.

Assumption of Positive Real Numbers

Assuming all variables represent positive real numbers ensures the principal square root is used and avoids complications with negative or complex values. This assumption justifies the direct application of radical multiplication rules without considering absolute values.