91Ó°ÊÓ

Identifying like terms is an essential skill when simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power, even if the coefficients—the numbers in front of the variables—are different. For instance, in the expression 2x + 3x, the terms 2x and 3x are considered like terms since both contain the variable x to the first power.

However, like terms can also involve radicals, including cube roots. In our exercise, the terms 9 \( \sqrt[3]{11} \) and 3 \( \sqrt[3]{11} \) have the identical radical part, which makes them like terms. This allows us to combine the terms by adding the coefficients while retaining the common radical part untouched.

Understanding like terms helps streamline the simplification process by consolidating and reducing an expression to its simplest form.

The cube root of a number is a special value that, when multiplied by itself three times (cubed), gives that number. Represented by the symbol \(\sqrt[3]{\cdot}\), it essentially 'reverses' the cubing process. For example, if we have \(8\), its cube root is \(2\) because \(2 \cdot 2 \cdot 2 = 8\).

In our example, the cube root of 11 is expressed as \(\sqrt[3]{11}\). This is an irrational number because 11 is not a perfect cube. However, in the context of an algebraic expression like our exercise, it behaves just like any other algebraic term and can be part of 'like terms' if the radical component matches across terms.

When simplifying expressions that contain like terms, adding coefficients is a key step. Coefficients are the numerical part of the terms that stipulate how many 'units' of the variable or expression you have. In our context, the terms 9 \( \sqrt[3]{11} \) and 3 \( \sqrt[3]{11} \) have coefficients of 9 and 3, respectively.

To combine the terms, we simply add the coefficients: 9 + 3 = 12, while the cube root of 11 remains unchanged. Hence, the simplified form would be 12 \(\sqrt[3]{11}\). It's a straightforward process similar to adding whole numbers but in the context of algebraic expressions, it's crucial to verify that before you combine coefficients, the terms are indeed like terms.