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In mathematics, a cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 8 is 2, because 2 multiplied by itself three times (2 * 2 * 2) equals 8.

This concept is denoted as \( \sqrt[3]{x} \) for any number \( x \). It's similar to square roots but involves finding a number that multiplies into the original number three times, instead of two.

Cube roots are particularly useful when dealing with cubic equations or volumetric calculations. They can be simplified by prime factorization, which involves breaking a number down into its basic prime components.

Prime factorization is a way of expressing a number as the product of its prime numbers. Each number can be broken down this way, allowing us to simplify complex expressions such as those involving cube roots.

In the exercise, for \( \sqrt[3]{32} \), the number 32 is expressed as \( 2^5 \). Breaking 32 into prime factors means identifying that it's the product of five 2s. So, \( \sqrt[3]{32} \) simplifies to \( 2^{5/3} = 2 \times \sqrt[3]{4} \).

Similarly, for \( \sqrt[3]{108} \), 108 can be broken down into two 2s and three 3s: \( 2^2 \times 3^3 \). This simplification results in \( 3 \times \sqrt[3]{4} \).

Using prime factorization is particularly helpful with cube roots or any other roots, as it allows for easier simplification by identifying blocks of like terms.

When simplifying expressions, combining like terms is crucial. Like terms refer to terms in an expression that have identical variable parts raised to the same power, allowing them to be combined.

In the given exercise, after simplifying the cube roots, you end up with terms like \( 2\sqrt[3]{4} \) and \( 3\sqrt[3]{4} \). Since both terms share \( \sqrt[3]{4} \) as a common factor, they are considered like terms, which means they can be combined.

To combine these, subtract the coefficients: \( 2\sqrt[3]{4} - 3\sqrt[3]{4} = -\sqrt[3]{4} \).

Remember that combining like terms helps to simplify expressions into a more manageable form, making it easier to handle complex polynomial or radical equations. This step is vital in streamlining the math problem to reach the simplest expression.