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.____√√x²y

Verified step by step guidance

1. Start by rewriting the expression using rational exponents. The expression \( \sqrt{\sqrt{x^2 y}} \) can be rewritten as \( (x^2 y)^{1/4} \).
2. Apply the property of exponents \((a^m b^n)^p = a^{mp} b^{np}\) to distribute the exponent \(1/4\) to both \(x^2\) and \(y\).
3. This gives \(x^{2 \cdot 1/4} y^{1 \cdot 1/4}\).
4. Simplify the exponents: \(x^{1/2} y^{1/4}\).
5. Convert the expression back to radical notation: \(\sqrt{x} \cdot \sqrt[4]{y}\).

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 by converting them into exponent form, making it easier to manipulate algebraically.

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 √x represents the square root of x. Understanding how to convert between radical and exponent notation is crucial for simplifying expressions effectively.

Properties of Exponents

The properties of exponents are rules that govern how to manipulate expressions involving exponents. Key properties include the product of powers, quotient of powers, and power of a power. These rules are essential for simplifying expressions with rational exponents, as they provide a systematic approach to combining and reducing terms.