Use the rules for radicals to perform the indicated operations. Assume all variable expressions represent positive real numbers. ⁵√x² • ⁵√x³

Verified step by step guidance

1. Identify the operation: You are multiplying two fifth roots, \(\sqrt[5]{x^2}\) and \(\sqrt[5]{x^3}\).
2. Recall the property of radicals that allows multiplication under the same root: \(\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a \cdot b}\).
3. Apply this property to combine the radicals: \(\sqrt[5]{x^2} \cdot \sqrt[5]{x^3} = \sqrt[5]{x^2 \cdot x^3}\).
4. Use the laws of exponents to multiply inside the radical: \(x^2 \cdot x^3 = x^{2+3} = x^5\).
5. Rewrite the expression as \(\sqrt[5]{x^5}\) and recognize that the fifth root of \(x^5\) simplifies to \(x\) because \(\sqrt[5]{x^5} = x^{5/5} = x\).

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 multiplication under a single radical. For example, ⁿ√a × ⁿ√b = ⁿ√(a × b), which simplifies operations involving radicals with the same index.

Multiplication of Radicals with the Same Index

When multiplying radicals that have the same root index, you can combine them into one radical by multiplying their radicands. For instance, ⁵√x² × ⁵√x³ = ⁵√(x² × x³) = ⁵√x⁵, simplifying the expression efficiently.

Simplifying Radicals Using Exponent Rules

Radicals can be expressed as fractional exponents, where ⁿ√x^m = x^(m/n). This allows simplification by applying exponent rules, such as adding exponents when multiplying like bases, making it easier to simplify expressions like ⁵√x⁵ = x^(5/5) = x.