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Prime factorization is breaking down a composite number into the product of its prime factors. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The process is essential in simplifying radicals because it reveals the underlying structure of the number, which can be manipulated to simplify radical expressions.

For instance, consider the number 54 from the exercise. To perform prime factorization, you would divide 54 by the smallest prime number that goes into it and continue the process with the quotient, until all the resulting factors are prime. In this case, the prime factors of 54 are 2 and 3, resulting in the prime factorization of 54 being expressed as \(2 \times 3^{3}\). This is the first step in simplifying the given radical expression and is a skill that is essential in many areas of algebra, including working with radicals.

Radicals and roots relate to the operation of finding the 'root' of a number. In our example, we're dealing with cube roots, as indicated by the radical expression \(\sqrt[3]{\dots}\). Understanding how to manipulate radicals is critical in simplifying them. When a radical involves a perfect cube, for example, you can remove it from under the radical symbol by determining what number multiplied by itself three times gives the original number.

In the exercise, \(3^{3}\) and \(x^{3}\) are both perfect cubes. They can be 'taken out' of the radical, as their cube roots are simply 3 and x, respectively. This is why we can simplify the given expression to \(3x\sqrt[3]{2x^{2}}\). The ability to identify and extract perfect powers from under radical symbols is a valuable technique for simplifying such expressions.

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like \(x\) or \(y\)), and operations (such as addition, subtraction, multiplication, and division). The beauty of algebraic expressions lies in their use to represent complex relationships in a condensed form and the power to manipulate these expressions according to algebraic rules and properties.

For example, in the expression \(\sqrt[3]{54 x^{5}}\), we have a variable \(x\) raised to a power, combined with a radical. As shown in the solution, by applying knowledge of prime factorization and the properties of radicals, we can transform and simplify this expression. The ability to work with algebraic expressions, change their form, and simplify them allows for solving a wide range of problems in mathematics and applications in other fields. The provided exercise helps students to practice and get comfortable with manipulating algebraic expressions which involve radicals.