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In algebraic expressions, like terms are terms that have the exact same variables raised to the same powers. What makes them 'like' is that they can be combined using arithmetic operations such as addition or subtraction. Consider the terms \( 4 \sqrt[3]{6} \) and \( 3 \sqrt[3]{6} \); they are like terms because they both involve the cube root of 6 without any variable attached to it.

When simplifying an expression, identifying and combining these like terms is the first step. For instance, since \( 4 \sqrt[3]{6} \) and \( 3 \sqrt[3]{6} \) are like terms, you can simply add the coefficients (4 and 3 respectively) to get a single term, \( 7 \sqrt[3]{6} \). The same process applies to any terms with the same variables like \( 2 \sqrt[3]{x^{2}} \) and \( 6 \sqrt[3]{x^{2}} \), which can be combined to \( 4 \sqrt[3]{x^{2}} \).

The arithmetic laws, or properties of operations, are fundamental to working with algebraic expressions. These include the commutative, associative, and distributive laws. In the context of the given exercise, we mainly use the associative law of addition, which allows us to group like terms together and combine them.

For instance, the associative property states that \( (a + b) + c = a + (b + c) \) for any numbers a, b, and c. In practical terms, you can rearrange the terms and group them without changing the result. This is particularly helpful when simplifying algebraic expressions with like terms, as we can add or subtract them in any order to streamline the expression into fewer terms.

A cube root of a number x is a number y such that \( y^3 = x \). Symbolically, it is represented as \( \sqrt[3]{x} \). Cube roots can be both positive and negative because cubing a negative number still gives a negative result. For example, the cube root of 8 is 2, because \( 2^3 = 8 \), while the cube root of -27 is -3, since \( (-3)^3 = -27 \).

In algebraic simplification, understanding cube roots is vital when they are part of the terms in an expression. In the provided exercise, cube roots are treated as part of the terms, and like terms that include cube roots can be combined. This is why \( 4 \sqrt[3]{6} \) and \( 3 \sqrt[3]{6} \) are added together, their being cube roots doesn't affect the property that allows us to add them directly.

Performing algebraic operations involves working with variables and constants to manipulate algebraic expressions through addition, subtraction, multiplication, or division. The key to success with these operations is understanding the kinds of terms involved and how they interact based on the previously mentioned arithmetic laws.

In our exercise, the algebraic operations are focused on addition of like terms. Once the like terms are identified, we perform the addition operation by simply combining the coefficients. It is important to note that multiplication and division operations within an algebraic expression are subject to the same rules regarding like terms and arithmetic laws, but they apply to the relevant operation being performed.