Problem 150 - Elementary Algebra

Question

In the following exercises, simplify. $$ 11 \sqrt{7}-12 \sqrt{7} $$

Step-by-Step Solution

Answer

-\sqrt{7}

Step 1: Identify Like Terms

Both terms, \(11 \sqrt{7}\) and \(12 \sqrt{7}\), have the same radical part, \(\sqrt{7}\). This makes them like terms, which can be combined.

Step 2: Combine Like Terms

To combine like terms, subtract the coefficients of the terms. The coefficients are 11 and 12. Subtracting gives: \(11 - 12 = -1\).

Step 3: Attach the Radical Part

After combining the coefficients, attach the common radical part, \(\sqrt{7}\). Thus, \(11 \sqrt{7} - 12 \sqrt{7} = -1 \sqrt{7}\).

Key Concepts

like terms

When simplifying radical expressions, it is important to identify like terms. Like terms have the same radical part. In the exercise provided, both terms, 11\(\sqrt{7}\) and 12\(\sqrt{7}\), share the same radical part, \(\sqrt{7}\). Because they share this common part, we can combine them. This simplification process is similar to combining like terms in algebra where terms with the same variable get added or subtracted together. Identifying like terms is the first crucial step in making expressions more manageable.

coefficients

In a term like 11\(\sqrt{7}\), the number in front of the radical, 11, is called the coefficient. The coefficient represents how many times the radical is being considered. In our specific exercise, we have two terms: 11\(\sqrt{7}\) and 12\(\sqrt{7}\). Their coefficients are 11 and 12 respectively. When simplifying, we focus on these coefficients. For instance, think of it like subtracting apples: If you have 11 apples and take away 12 apples, mathematically it would give you \(-1\) apples. So, focusing on coefficients helps simplify radical expressions effectively.

subtraction of radicals

Once we identify like terms and understand coefficients, we move on to subtracting radicals. In the example, we subtract the coefficients of 11\(\sqrt{7}\) and 12\(\sqrt{7}\). This subtraction process is straightforward: 11 - 12 gives us -1. After subtracting the coefficients, we reattach the common radical part \(\sqrt{7}\). So, 11\(\sqrt{7}\) - 12\(\sqrt{7}\) simplifies to -1\(\sqrt{7}\). By understanding the subtraction of radicals, students can simplify expressions confidently and accurately.