Problem 33 - Algebra and Trigonometry

Question

Add or subtract terms whenever possible. \(7 \sqrt{3}+6 \sqrt{3}\)

Step-by-Step Solution

Answer

The answer is \(13 \sqrt{3}\). Such like terms can be combined by adding their coefficients.

Step 1: Identify Like Terms

In this sum \(7 \sqrt{3}+6 \sqrt{3}\), the terms \(7 \sqrt{3}\) and \(6 \sqrt{3}\) are like terms, because they consist of the same mathematical expressions, in this case \( \sqrt{3}\).

Step 2: Add the Coefficients of Like Terms

Add the coefficients of the two like terms, which are 7 and 6, together. This results in 13.

Step 3: Rewrite the Result

The result of adding the coefficients should be placed in front of the like term. This gives us \(13 \sqrt{3}\).

Key Concepts

Understanding Like Terms

In mathematics, terms are classified as "like terms" when they contain the same variable components raised to the same power. In the context of radicals, like terms share the same radical part. For instance, in the expression \( 7 \sqrt{3} + 6 \sqrt{3} \), both terms \( 7 \sqrt{3} \) and \( 6 \sqrt{3} \) are like terms because they include the radical \( \sqrt{3} \). The numerical values in front of these radicals, such as 7 and 6, do not affect whether the terms are like terms; only the radical expression does.
Recognizing like terms in an equation or expression enables us to simplify it by combining these terms. It's similar to adding apples to apples, where the only things we consider are the coefficients sitting in front. Look for the identical format or structure in terms to easily spot the ones that can be added together.

The Role of Coefficients

Coefficients are the numerical factors of terms in an expression. They sit directly in front of a variable or radical. In the expression \( 7 \sqrt{3} + 6 \sqrt{3} \), the coefficients are 7 and 6, which belong to their respective terms.

• The coefficients tell us how many times the base (radical or variable) is being counted.
• They can be easily added or subtracted when dealing with like terms since the radical part stays constant.

By focusing on the coefficients, we essentially simplify the process by only having to deal with adding the numbers, rather than the radicals themselves. Once summed, this new coefficient is combined again with the common radical part to form a simplified term.

Adding Radicals

Adding radicals is straightforward when you understand the concept of like terms and coefficients. For radicals to be added directly, they must be like terms, meaning each contains the same radical component. Let's revisit the example \( 7 \sqrt{3} + 6 \sqrt{3} \).

• First, ensure the radicals are identical; observe \( \sqrt{3} \) in both terms.
• Proceed by adding their coefficients: 7 and 6 to get a combined coefficient of 13.
• Finally, write the result as \( 13 \sqrt{3} \).

This simple addition of coefficients fulfills the criteria for adding radicals. Always keep the original radical the same in the computed answer. Remember, unlike adding regular numbers, adding radicals requires this matching of radical parts before performing the addition of coefficients.