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A cube root refers to a special value that, when used three times in multiplication, gives the original number. For instance, the cube root of 8 is 2 because multiplying 2 by itself three times (\[ 2 \times 2 \times 2 = 8 \]) equals 8. The cube root is represented using the radical symbol with a small three (index) above it, like this: \( \sqrt[3]{x} \). This signifies the cube root of a number \(x\). In the context of our exercise, \( \sqrt[3]{3} \) denotes the cube root of 3.

Understanding cube roots helps in simplifying and solving radical expressions, especially when combining or separating them. Remember, only like terms—which we’ll discuss later—can be added or subtracted effectively when dealing with cube roots and other radicals.

When you come across radical expressions, adding or subtracting them is quite similar to combining like terms in algebra. However, one critical condition must be met: the radicals involved must be the same.

This means that the expression under the radical sign (the radicand) must be identical. For example, you can add or subtract \( 4 \sqrt[3]{5} \) and \( 2 \sqrt[3]{5} \) because both have \( \sqrt[3]{5} \) as the radical part. But you cannot directly add \( 4 \sqrt[3]{5} \) to \( 2 \sqrt[3]{3} \) because their radicands differ.

• If the radicals are not the same, try simplifying them first. Sometimes, this simplification can reveal like terms.
• When adding or subtracting, focus on the coefficients—the numbers in front of the radicals—just like in our example from the exercise.

By understanding these rules, you can easily manipulate and combine radical expressions, simplifying problems significantly.

In algebra, 'like terms' refer to terms that have identical variable parts raised to the same power. This concept extends to radicals, where terms are "like" if their radicands and index numbers match. For example, \( 7 \sqrt[3]{a} \) and \( -3 \sqrt[3]{a} \) are like terms because they share the same cube root of \(a\).

The importance of identifying like terms becomes clear when performing operations such as addition or subtraction simply because you can consolidate them into a single term.

• The coefficients, which are the numbers in front of the radicals, are combined just as you would combine ordinary like terms in an algebraic expression.
• This simplifies expressions greatly, making it easier to understand and manage them.

Consider the original exercise \( 6 \sqrt[3]{3} - 2 \sqrt[3]{3} \). Both terms have \( \sqrt[3]{3} \) as their radical part, signaling that they are like terms. Thus, you simply subtract their coefficients, leading to \( 4 \sqrt[3]{3} \). Recognizing like terms allows for efficient and accurate mathematical operations.