In Exercises 39–60, simplify by factoring. Assume that all variables in a radicand represent positive real numbers and no radicands involve negative quantities raised to even powers. _____ ³√x³y¹⁷z²

Verified step by step guidance

1. Identify the expression under the cube root: $x^3{y}^{17}{z}^2$.
2. Recognize that the cube root of a product is the product of the cube roots: $\sqrt[3]{x^3{y}^{17}{z}^2}=\sqrt[3]{x^3}\cdot \sqrt[3]{y^{17}}\cdot \sqrt[3]{z^2}$.
3. Simplify each part separately: $\sqrt[3]{x^3}=x$ because $x^3$ is a perfect cube.
4. For $\sqrt[3]{y^{17}}$, factor $y^{17}$ as $y^{15}\cdot y^2$ and simplify: $\sqrt[3]{y^{15}}=y^5$ and $\sqrt[3]{y^2}$ remains under the cube root.
5. For $\sqrt[3]{z^2}$, it remains under the cube root as $z^2$ is not a perfect cube.

Key Concepts

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. In algebra, this often involves identifying common factors or applying specific techniques such as grouping or using special products. Understanding how to factor is essential for simplifying expressions, especially when dealing with polynomials or radical expressions.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots, and are represented using the radical symbol (√). In the context of the given question, the cube root (³√) indicates that we are looking for factors of the expression that can be expressed as perfect cubes. Simplifying radical expressions often requires recognizing and extracting these perfect powers from under the radical.

Properties of Exponents

The properties of exponents are rules that govern how to manipulate expressions involving powers. Key properties include the product of powers, power of a power, and the quotient of powers. These rules are crucial when simplifying expressions like ³√x³y¹⁷z², as they allow us to rewrite the expression in a more manageable form by applying the appropriate exponent rules to each variable.