Problem 15 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ 11 \sqrt{3}-12 \sqrt{3} $$

Step-by-Step Solution

Answer

-\(\sqrt{3}\).

Step 1: Identify like terms

In this expression, \(11 \sqrt{3}\) and \(-12 \sqrt{3}\) are like terms because they both contain the square root of 3.

Step 2: Apply the subtraction

Subtract \(12 \sqrt{3}\) from \(11 \sqrt{3}\) taking into account the negative sign. This operation will result in the simplification of the expression.

Key Concepts

Like Terms

When working with radical expressions, identifying like terms is crucial. Like terms are terms that have exactly the same radical part. In our example, both terms have \( \sqrt{3} \) as their radical component, making them like terms.

To determine like terms, focus on the variable and the exponent parts, if any, rather than the coefficients. In radical expressions, the radical part inside the square root is what you look for. Any difference in this part means the terms are not like.

This concept is similar to combining regular algebraic terms such as \( 2x \) and \( 5x \), where the variable \( x \) makes them like terms. But with radicals, check whether the numbers under the square root are identical.

Subtraction of Radicals

Subtraction of radicals works very much like the subtraction of regular numbers, provided they are like terms. Think of subtracting radicals as you would with regular algebraic terms.

In the exercise \(11 \sqrt{3} - 12 \sqrt{3}\), since both expressions include \(\sqrt{3}\), subtraction can proceed as if you're just handling the coefficients: 11 and -12. You subtract one coefficient from the other while keeping the radical part the same.

One possible misunderstanding might be to treat \(11 \sqrt{3}\) and \(-12 \sqrt{3}\) as unrelated terms due to their negative or positive signs. Remember to focus on the radical when determining if they are "subtractable" or not.

Negative Numbers

Understanding negative numbers is fundamental when simplifying expressions. Negative signs can change the entire value of a term, including radicals. In expressions like \(-12 \sqrt{3}\), the negative sign applies to the whole term coefficients, thus transforming subtraction into an action that affects the value in opposite directions.

When you're subtracting a larger radical from a smaller one, the result is negative, as in the expression \(11 \sqrt{3} - 12 \sqrt{3}\). The subtraction results in \(-1 \sqrt{3}\).

To avoid confusion, it's helpful to break it down into steps:

• Look at the first term: 11 times \(\sqrt{3}\)
• Observe the second term: -12 times \(\sqrt{3}\)
• Perform the subtraction: 11 - 12 = -1 while retaining \(\sqrt{3}\)

The negative result could mean that there is a deeper context in solving larger problems, often indicating a direction change or deficit.