Perform the indicated operations. Assume all variables represent positive real numbers. 3∜x⁵y - 2x∜xy

Verified step by step guidance

1. Identify the terms in the expression: \(3\sqrt[4]{x^5y}\) and \(-2x\sqrt[4]{xy}\).
2. Simplify each term separately. For \(3\sqrt[4]{x^5y}\), rewrite it as \(3(x^5y)^{1/4}\).
3. For \(-2x\sqrt[4]{xy}\), rewrite it as \(-2x(xy)^{1/4}\).
4. Apply the property of exponents: \((a^m b^n)^p = a^{mp} b^{np}\). Simplify \((x^5y)^{1/4}\) to \(x^{5/4}y^{1/4}\) and \((xy)^{1/4}\) to \(x^{1/4}y^{1/4}\).
5. Combine like terms if possible by factoring out common terms from the simplified expressions.

Key Concepts

Radicals and Rational Exponents

Radicals, such as square roots and cube roots, can be expressed using rational exponents. For example, the cube root of a number 'a' can be written as a^(1/3). Understanding how to manipulate these expressions is crucial for performing operations involving roots and exponents.

Combining Like Terms

Combining like terms involves simplifying expressions by adding or subtracting terms that have the same variable raised to the same power. This is essential in algebra to reduce expressions to their simplest form, making it easier to perform further operations.

Factoring and Distributing

Factoring is the process of breaking down an expression into simpler components, while distributing involves multiplying a term across a sum or difference. Both techniques are vital for simplifying algebraic expressions and solving equations, especially when dealing with polynomials and radicals.