In Exercises 1–20, use the product rule to multiply.___ __⁴√6x² ⋅ ⁴√3x

Verified step by step guidance

1. Identify the expression: \( \sqrt[4]{6x^2} \cdot \sqrt[4]{3x} \).
2. Use the property of radicals: \( \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b} \).
3. Combine the expressions under a single fourth root: \( \sqrt[4]{(6x^2) \cdot (3x)} \).
4. Multiply the terms inside the radical: \( 6x^2 \cdot 3x = 18x^3 \).
5. Express the result as a single radical: \( \sqrt[4]{18x^3} \).

Key Concepts

Product Rule

The product rule is a fundamental property of exponents that states when multiplying two expressions with the same root, you can combine them under a single root. Specifically, for roots, this means that √a ⋅ √b = √(a*b). This rule simplifies the multiplication of radical expressions, making it easier to work with them.

Radical Expressions

Radical expressions involve roots, such as square roots or fourth roots, and are written in the form √a or n√a, where 'n' indicates the degree of the root. Understanding how to manipulate these expressions, including simplifying and combining them, is crucial for solving problems involving radicals.

Simplifying Radicals

Simplifying radicals involves reducing the expression to its simplest form, which often includes factoring out perfect squares or higher powers from under the root. This process not only makes calculations easier but also helps in understanding the properties of the numbers involved, leading to clearer solutions in algebraic problems.