In Exercises 11–28, add or subtract as indicated. You will need to simplify terms to identify the like radicals. ___ ___ 4³√x⁴y² + 5x³√xy²

Verified step by step guidance

1. Identify the like radicals in the expression: $4\sqrt[3]{x^4{y}^2}+5x\sqrt[3]{x{y}^2}$.
2. Simplify each term to see if they can be combined. Start with $4\sqrt[3]{x^4{y}^2}$:
3. Rewrite $x^4$ as $x^3\cdot x$ and $y^2$ as $y^2$ to simplify the radical: $4\sqrt[3]{x^3\cdot x\cdot y^2}=4x\sqrt[3]{x{y}^2}$.
4. Now simplify the second term $5x\sqrt[3]{x{y}^2}$. It is already in its simplest form.
5. Since both terms are now $x\sqrt[3]{x{y}^2}$, combine them: $(4+5)x\sqrt[3]{x{y}^2}$.

Key Concepts

Radicals

Radicals are expressions that involve roots, such as square roots or cube roots. In the context of algebra, simplifying radicals involves rewriting them in a simpler form, often by factoring out perfect squares or cubes. Understanding how to manipulate radicals is essential for performing operations like addition and subtraction.

Like Radicals

Like radicals are terms that have the same root and radicand (the expression inside the radical). For example, 3√2 and 5√2 are like radicals because they both contain √2. When adding or subtracting like radicals, you combine their coefficients while keeping the radical part unchanged, similar to combining like terms in polynomial expressions.

Simplification of Expressions

Simplification of expressions involves reducing them to their simplest form, which often includes combining like terms and reducing fractions. In the case of radical expressions, this means identifying and simplifying terms that can be combined, ensuring that the final expression is as concise and clear as possible. This process is crucial for accurately performing operations such as addition and subtraction.