Problem 31 - Algebra A Combined Function

Question

Add or subtract as indicated. Assume that all variables represent positive real numbers. $$ 6 \sqrt[3]{11}+8 \sqrt{11}-12 \sqrt{11} $$

Step-by-Step Solution

Answer

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

Step 1: Identify Like Terms

To simplify the expression, first look for terms that have the same radical part. The expression is \( 6 \sqrt[3]{11} + 8 \sqrt{11} - 12 \sqrt{11} \). The terms \( 8 \sqrt{11} \) and \( -12 \sqrt{11} \) both have the same radical part \( \sqrt{11} \).

Step 2: Combine Like Terms

Combine the like terms, \( 8 \sqrt{11} \) and \( -12 \sqrt{11} \). This involves performing the arithmetic operation on the coefficients: \( 8 - 12 = -4 \). So, \( 8 \sqrt{11} - 12 \sqrt{11} = -4 \sqrt{11} \).

Step 3: Write Final Expression

The final simplified expression involves writing the terms with different radicals unchanged and combined results together. There is no change to the \( 6 \sqrt[3]{11} \) term. Therefore, the expression simplifies to \( 6 \sqrt[3]{11} - 4 \sqrt{11} \).

Key Concepts

Like Terms

When simplifying expressions, identifying like terms is essential. Like terms are terms in an expression that have the exact same variables raised to the same power. In the context of radical expressions, like terms share the same radical part or root.

• Example: In the expression \(6 \sqrt[3]{11} + 8 \sqrt{11} - 12 \sqrt{11}\), the terms \(8 \sqrt{11}\) and \(-12 \sqrt{11}\) are considered like terms because they both have \(\sqrt{11}\) as their radical part.
• Importance: Identifying like terms allows you to combine them into a single term, which simplifies the expression.

To combine them, perform the algebraic operation (addition or subtraction) on their coefficients while keeping the radical part the same. This leads to a cleaner and more manageable expression.

Radicals

Radicals are symbols used to represent the root of a number. The most common radicals are square roots and cube roots.

• Square Root (\(\sqrt{}\)): The square root of a number \(x\) is a value that, when multiplied by itself, gives \(x\).
• Cube Root (\(\sqrt[3]{}\)): The cube root of a number \(x\) is a value that, when raised to the power of three, results in \(x\).

In the original exercise, both square roots \(\sqrt{11}\) and cube roots \(\sqrt[3]{11}\) are present. They indicate different operations, affecting how we can combine terms. Terms with different radicals cannot be combined as like terms. So, \(6 \sqrt[3]{11}\) remains separate from any terms involving \(\sqrt{11}\). Understanding this distinction is crucial when simplifying radical expressions.

Coefficients

Coefficients are the numerical parts that multiply the variable or radical part in a term. In the expression \(6 \sqrt[3]{11} + 8 \sqrt{11} - 12 \sqrt{11}\), numbers like 6, 8, and -12 are coefficients.

• Function: Coefficients determine the size or the magnitude of a term. They adjust the impact each term has on the total expression.
• Combining Terms: When working with like terms, you only combine the coefficients. For example, to simplify \(8 \sqrt{11} - 12 \sqrt{11}\), focus on the coefficients: \(8 - 12 = -4\).

This results in \(-4 \sqrt{11}\). By treating coefficients independently when simplifying, you are able to maintain the properties of the radicals unchanged. This understanding ensures that any combining of terms or simplification respects the algebraic structure of the expression.