91Ó°ÊÓ

In mathematics, 'like terms' are terms that have the same variables raised to the same power.
These are essential in simplifying expressions or solving equations because they can be easily combined.
In the expression provided, we look for like terms, which involves identifying terms with the same cube root.
The terms given are \(9\sqrt[3]{7}\), \(3\sqrt[3]{7}\), and \(-4\sqrt[3]{7}\).
All of these terms include the cube root of 7, making them like terms:

• The variable component, \(\sqrt[3]{7}\), is identical in each term.
• This allows us to focus solely on combining their coefficients.

Recognizing like terms simplifies the process of combining them because we only need the numerical part for calculations.
By concentrating on these coefficients, we can streamline expressions easily and efficiently.

When working with like terms, the next step is combining coefficients.
Coefficients are simply the numerical part of the terms—those numbers directly in front of the variable component.
For our exercise, we have the following coefficients: 9, 3, and -4. Combining these coefficients follows basic arithmetic:

• Add together the positive coefficients: 9 + 3 = 12.
• Then subtract the negative coefficient: 12 - 4 = 8.

This gives us a total of 8, which represents the combined coefficient of our like terms.
Once combined, this coefficient is used with the like terms to express the simplified form of the original expression.
The power of combining coefficients lies in letting us manage otherwise complex expressions by reducing them to more straightforward math tasks.

The cube root is a specialized function in mathematics, denoted by the symbol \(\sqrt[3]{}\).
It's the operation of finding a number which, when multiplied by itself twice, returns the original number.
For instance, in our problem the cube root of 7, \(\sqrt[3]{7}\), can be understood as:

• A number that—if multiplied three times—results in 7.
• It is analogous to the square root but involves an additional multiplication.

This function is crucial when dealing with certain algebraic expressions, often simplifying into like terms fitting into broader equations easily.
In our exercise, cube roots facilitate grouping terms under a common factor, ensuring mathematical coherence when simplifying the expression.
Understanding cube roots and their properties is fundamental for simplifying complex expressions—and integral in various mathematical fields and applications.