In Exercises 55–78, use properties of rational exponents to simplify each expression. Assume that all variables represent positive numbers.(x^½y^-⅗)^½

Verified step by step guidance

1. Start by applying the power of a power property: \((a^m)^n = a^{m \cdot n}\).
2. Apply this property to each part of the expression \((x^{1/2}y^{-3/5})^{1/2}\).
3. Simplify the exponent of \(x\): \((x^{1/2})^{1/2} = x^{(1/2) \cdot (1/2)} = x^{1/4}\).
4. Simplify the exponent of \(y\): \((y^{-3/5})^{1/2} = y^{(-3/5) \cdot (1/2)} = y^{-3/10}\).
5. Combine the simplified expressions: \(x^{1/4}y^{-3/10}\).

Key Concepts

Rational Exponents

Rational exponents are a way to express roots and powers using fractions. For example, an exponent of 1/2 represents the square root, while an exponent of -3/5 indicates both a root and a reciprocal. Understanding how to manipulate these exponents is crucial for simplifying expressions involving them.

Properties of Exponents

The properties of exponents include rules such as the product of powers, power of a power, and quotient of powers. These rules allow us to combine and simplify expressions with exponents effectively. For instance, when raising a power to another power, you multiply the exponents, which is essential for simplifying the given expression.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, often by combining like terms and applying exponent rules. This process is vital in algebra as it makes expressions easier to work with and understand. In the context of the given problem, applying the properties of rational exponents will help in achieving a more manageable expression.