Simplify each radical. Assume all variables represent positive real numbers. $\sqrt[6]{x^{18}y^2}$

Verified step by step guidance

1. Identify the expression inside the sixth root: \(\sqrt[6]{x^{18} y^{2}}\).
2. Recall the property of radicals that \(\sqrt[n]{a^{m}} = a^{\frac{m}{n}}\). Apply this to each variable inside the radical: \(\sqrt[6]{x^{18}} = x^{\frac{18}{6}}\) and \(\sqrt[6]{y^{2}} = y^{\frac{2}{6}}\).
3. Simplify the exponents by dividing: \(x^{\frac{18}{6}} = x^{3}\) and \(y^{\frac{2}{6}} = y^{\frac{1}{3}}\).
4. Rewrite the expression as a product of powers: \(x^{3} y^{\frac{1}{3}}\).
5. Express the fractional exponent back as a radical if desired: \(y^{\frac{1}{3}} = \sqrt[3]{y}\). So the simplified form is \(x^{3} \sqrt[3]{y}\).

Key Concepts

Properties of Radicals

Radicals represent roots of numbers or expressions. The nth root of a product equals the product of the nth roots, allowing us to separate and simplify each factor under the radical individually.

Exponents and Roots Relationship

An nth root can be expressed as a fractional exponent 1/n. For example, the sixth root of x¹⁸ is x raised to the power 18/6, which simplifies the expression by converting roots into exponents.

Simplifying Expressions with Variables

When simplifying radicals with variables, divide the exponent by the root index to find the simplified power. Since variables are positive, no absolute value is needed, and leftover exponents remain under the radical.