In Exercises 79–112, use rational exponents to simplify each expression. If rational exponents appear after simplifying, write the answer in radical notation. Assume that all variables represent positive numbers._____⁵√x¹⁰y¹⁵

Verified step by step guidance

1. Identify the expression given: \( \sqrt[5]{x^{10}y^{15}} \).
2. Rewrite the expression using rational exponents: \( (x^{10}y^{15})^{1/5} \).
3. Apply the power of a power property: \( x^{10 \cdot 1/5}y^{15 \cdot 1/5} \).
4. Simplify the exponents: \( x^{2}y^{3} \).
5. Convert back to radical notation if needed: \( x^{2}y^{3} \) is already in simplest form.

Key Concepts

Rational Exponents

Rational exponents are a way to express roots using fractional powers. For example, the expression x^(1/n) represents the nth root of x. This concept allows for the simplification of expressions involving roots and powers, making calculations more manageable. Understanding how to convert between radical and exponent notation is crucial for solving problems involving rational exponents.

Radical Notation

Radical notation is a mathematical notation used to denote roots, such as square roots or cube roots. The radical symbol (√) indicates the root of a number, where the index of the root is specified when it is not a square root. Converting expressions from rational exponents to radical notation is often required in algebra to present answers in a more recognizable form, especially when dealing with roots of variables.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, which often includes combining like terms, reducing fractions, and applying exponent rules. In the context of rational exponents, this means using properties of exponents to rewrite expressions in a more manageable way. Mastery of simplification techniques is essential for effectively solving algebraic problems and ensuring accurate results.