Perform the indicated operations. Assume all variables represent positive real numbers. $3\sqrt[4]{x^5y} - 2x\sqrt[4]{xy}$

Verified step by step guidance

1. Rewrite the given expression by expressing the radicals with fractional exponents. Recall that the fourth root of a variable is the variable raised to the power of \( \frac{1}{4} \). So, \( \sqrt[4]{x} = x^{\frac{1}{4}} \) and \( \sqrt[4]{xy} = (xy)^{\frac{1}{4}} \).
2. Express each term using fractional exponents: First term: \( 3 \sqrt[4]{x^{5} y} = 3 (x^{5} y)^{\frac{1}{4}} = 3 x^{\frac{5}{4}} y^{\frac{1}{4}} \) Second term: \( 2 x \sqrt[4]{x y} = 2 x (x y)^{\frac{1}{4}} = 2 x \cdot x^{\frac{1}{4}} y^{\frac{1}{4}} = 2 x^{1 + \frac{1}{4}} y^{\frac{1}{4}} = 2 x^{\frac{5}{4}} y^{\frac{1}{4}} \).
3. Now, rewrite the expression as: \( 3 x^{\frac{5}{4}} y^{\frac{1}{4}} - 2 x^{\frac{5}{4}} y^{\frac{1}{4}} \).
4. Since both terms have the same variables with the same exponents, factor out the common term \( x^{\frac{5}{4}} y^{\frac{1}{4}} \): \( (3 - 2) x^{\frac{5}{4}} y^{\frac{1}{4}} \).
5. Simplify the coefficients inside the parentheses to get the simplified expression in terms of fractional exponents or, if preferred, rewrite back to radical form.

Key Concepts

Radical Expressions and Roots

Radical expressions involve roots such as square roots, cube roots, and fourth roots. Understanding how to interpret and simplify expressions with radicals, like the fourth root (∜), is essential. For example, ∜x means the fourth root of x, which is x raised to the power of 1/4.

Properties of Exponents

Exponents rules allow us to simplify expressions involving powers and roots by rewriting radicals as fractional exponents. For instance, ∜x⁵ can be written as x^(5/4). Knowing how to add, subtract, multiply, and divide expressions with exponents is key to combining like terms.

Combining Like Terms with Radicals

To perform operations like addition or subtraction on radical expressions, the terms must have the same radical part (like terms). This means the variables and their exponents under the root must match. Simplifying each term to a common radical form helps identify like terms for combination.