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Like terms are terms in an expression that share the same variables raised to the same powers. In this context, when we look at radicals, we consider them to be parts of a term similar to variables. When simplifying expressions, identifying like terms is crucial because it allows us to combine them to make the expression more streamlined.

To identify like terms with radicals, check if the radical part of each term is the same. If they share the same radical, you can think of them as similar, much like how you would treat terms such as \(2x\) and \(3x\) as like terms because they both have \(x\).

In the exercise, terms like \(3\sqrt{3}\), \(8\sqrt{3}\), and \(-16\sqrt{3}\) are like terms because they all have \(\sqrt{3}\). This shared radical lets you combine their coefficients effectively.

Radicals are symbols that represent the root of a number. The most common radical is the square root, denoted by \(\sqrt{}\). This symbol asks for the number that, when multiplied by itself, gives the original number under the radical sign.

Understanding radicals is key to simplifying expressions that include them. By treating radicals like variables in terms, we can combine coefficients to simplify the expression.

In the example, \(\sqrt{3}\) is the radical part of each term. Recognizing this allows you to focus on manipulating the numerical coefficients without changing the radical part itself.

Coefficients are the numerical parts of terms that multiply the variable or radical. When simplifying expressions, focusing on the coefficients allows you to combine terms easily.

In expressions with radicals as in the exercise, once you identify the like terms, you intently work with the coefficients. These numbers, such as 3, 8, and -16, are what you add or subtract.

By calculating \((3 + 8 - 16)\), you find the coefficient of the simplified expression, resulting in \(-5\). This allows you to write the final simplified term as \(-5\sqrt{3}\), effectively combining the original terms into a more compact form.