Problem 31 - Intermediate Algebra

Question

Add or subtract. $$ 6 \sqrt[3]{11}+8 \sqrt{11}-12 \sqrt{11} $$

Step-by-Step Solution

Answer

The simplified expression is \(6 \sqrt[3]{11} - 4 \sqrt{11}\).

Step 1: Identify Like Terms

In the expression \(6 \sqrt[3]{11} + 8 \sqrt{11} - 12 \sqrt{11}\), identify terms with the same root. \(6 \sqrt[3]{11}\) is a cube root term, while \(8 \sqrt{11}\) and \(12 \sqrt{11}\) are square root terms.

Step 2: Combine Like Terms

Combine the square root terms \(8 \sqrt{11}\) and \(-12 \sqrt{11}\). The coefficients are 8 and -12 respectively, so combining them gives \(8 - 12 = -4\). Thus, the expression becomes \(6 \sqrt[3]{11} - 4 \sqrt{11}\).

Step 3: Simplify the Expression

Since there are no more like terms that can be combined, the simplified form of the expression is \(6 \sqrt[3]{11} - 4 \sqrt{11}\).

Key Concepts

Like Terms

In algebra, like terms are terms that contain the same variables raised to the same power, making them directly comparable and combinable. For example, in the expression \(6 \sqrt[3]{11} + 8 \sqrt{11} - 12 \sqrt{11}\), identifying like terms involves finding terms with similar bases and operations, such as square roots or cube roots.

• Cube root term: \(6 \sqrt[3]{11}\)
• Square root terms: \(8 \sqrt{11}\) and \(12 \sqrt{11}\)

The square root terms \(8 \sqrt{11}\) and \(-12 \sqrt{11}\) are like terms because they share the same root value and operation \(\sqrt{11}\). Conversely, \(6 \sqrt[3]{11}\) is not a like term with the square root terms, since it's a cube root. Recognizing like terms is crucial because it allows us to execute operations like addition or subtraction on them, simplifying algebraic expressions efficiently.

Cube Root

The cube root function is the inverse of cubing a number. While the process of finding a cube, such as raising a number \(x\) to the power of three, is denoted by \(x^3\), the cube root finds a number that, when cubed, results in the original value. It is symbolized by the radical symbol \(\sqrt[3]{ } \).
For example, \(\sqrt[3]{8} = 2\) because \(2^3 = 8\). In the expression \(6 \sqrt[3]{11}\), the \(\sqrt[3]{11}\) refers to the cube root of 11. Cube roots are important in algebra for simplifying expressions, especially when these expressions involve higher-degree roots. Understanding cube roots is essential for solving complex problems and manipulating algebraic expressions that involve multiples of roots and powers.

Square Root

Square roots are one of the most frequently used roots in mathematics, representing the opposite of squaring a number. Symbolized by \(\sqrt{ }\), the square root of a number \(n\) is a value that yields \(n\) when multiplied by itself.
For example, \(\sqrt{25} = 5\) because \(5 \times 5 = 25\). In the given expression, we see square root terms like \(8 \sqrt{11}\) and \(-12 \sqrt{11}\). These can be considered like terms because they are both multiplied by the square root of the same number, 11.
The distinction between cube roots and square roots is important when identifying like terms. This helps in accurately combining terms in algebraic expressions.

Coefficients

Coefficients in algebra refer to the numerical component in front of a variable or radical in an expression. These numbers represent how many times the variable or set root is being considered. For example, in the terms \(8 \sqrt{11}\) and \(-12 \sqrt{11}\), 8 and -12 are the coefficients.
Coefficients play a key role when combining like terms, as they dictate how terms are added or subtracted. In our original exercise, to simplify \(8 \sqrt{11} - 12 \sqrt{11}\), we subtract the coefficients: \(8 - 12 = -4\).
Consequently, the resulting term becomes \(-4 \sqrt{11}\). Understanding coefficients allows for effective simplification and manipulation of algebraic expressions, ensuring clarity and precision in problem-solving.