Perform the indicated operations. Assume all variables represent positive real numbers. $\(\sqrt\)[4]{81x^6y^3} - \(\sqrt\)[4]{16x^{10}y^3}$

Verified step by step guidance

1. Rewrite each term using fractional exponents to express the fourth roots: \(\sqrt[4]{81x^{6}y^{3}} = (81x^{6}y^{3})^{\frac{1}{4}}\) and \(\sqrt[4]{16x^{10}y^{3}} = (16x^{10}y^{3})^{\frac{1}{4}}\).
2. Apply the exponent to each factor inside the parentheses separately using the property \((abc)^m = a^m b^m c^m\): \((81)^{\frac{1}{4}} (x^{6})^{\frac{1}{4}} (y^{3})^{\frac{1}{4}}\) and \((16)^{\frac{1}{4}} (x^{10})^{\frac{1}{4}} (y^{3})^{\frac{1}{4}}\).
3. Simplify the numerical parts by finding the fourth root of 81 and 16: \$81 = 3^4\(, so \((81)^{\frac{1}{4}} = 3\), and \)16 = 2^4$, so \((16)^{\frac{1}{4}} = 2\).
4. Simplify the variable parts by multiplying the exponents: \((x^{6})^{\frac{1}{4}} = x^{\frac{6}{4}} = x^{\frac{3}{2}}\), \((x^{10})^{\frac{1}{4}} = x^{\frac{10}{4}} = x^{\frac{5}{2}}\), and \((y^{3})^{\frac{1}{4}} = y^{\frac{3}{4}}\) for both terms.
5. Rewrite the expression with the simplified parts and then perform the subtraction: \(3 x^{\frac{3}{2}} y^{\frac{3}{4}} - 2 x^{\frac{5}{2}} y^{\frac{3}{4}}\). From here, factor out the common terms if possible to simplify further.

Key Concepts

Radical Expressions and Roots

Radical expressions involve roots such as square roots, cube roots, and fourth roots (∜). The fourth root of a number is the value that, when raised to the fourth power, gives the original number. Understanding how to simplify and manipulate these roots is essential for performing operations on radical expressions.

Properties of Exponents

Exponents indicate repeated multiplication, and their properties help simplify expressions involving powers. When dealing with roots, exponents can be rewritten as fractional powers (e.g., the fourth root corresponds to the exponent 1/4). Applying exponent rules allows for simplification and combination of terms under radicals.

Operations with Like Radicals

To add or subtract radical expressions, the radicals must be like terms, meaning they have the same index and radicand. Simplifying each radical to its simplest form helps identify like radicals. Once like radicals are identified, their coefficients can be combined through addition or subtraction.