Perform the indicated operations. Assume all variables represent positive real numbers. ∜81x⁶y³ - ∜16x¹⁰y³

Verified step by step guidance

1. Identify the expression: \( \sqrt[4]{81x^6y^3} - \sqrt[4]{16x^{10}y^3} \).
2. Break down each radical expression into its prime factors: \( 81 = 3^4 \), \( 16 = 2^4 \), and express the variables with exponents divisible by 4.
3. Rewrite the expression using the properties of exponents: \( \sqrt[4]{3^4(x^6)(y^3)} - \sqrt[4]{2^4(x^{10})(y^3)} \).
4. Simplify each term by taking the fourth root of the perfect powers: \( 3x^{1.5}y^{0.75} - 2x^{2.5}y^{0.75} \).
5. Factor out the common term from both expressions: \( y^{0.75}(3x^{1.5} - 2x^{2.5}) \).

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots or fourth roots, which are used to express numbers in terms of their base and exponent. In this question, the fourth root (∜) is applied to the expressions 81x⁶y³ and 16x¹⁰y³. Understanding how to simplify and manipulate these expressions is crucial for performing the indicated operations.

Properties of Exponents

The properties of exponents govern how to handle expressions involving powers. Key rules include the product of powers, quotient of powers, and power of a power. In this problem, recognizing how to apply these properties when simplifying the terms after taking the fourth root is essential for accurate calculations.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, which can include combining like terms and factoring. In the context of this question, after calculating the fourth roots, it is important to simplify the resulting expressions to arrive at a final answer. This skill is fundamental in algebra for clarity and efficiency in problem-solving.