Problem 161 - Elementary Algebra

Question

In the following exercises, simplify. $$ 8 \sqrt{7}+2 \sqrt{7}+3 \sqrt{7} $$

Step-by-Step Solution

Answer

13 \(\sqrt{7}\)

Step 1 - Identify Like Terms

In the expression, identify the like terms. Here, all the terms involve a square root of 7: \(8 \sqrt{7}\), \(2 \sqrt{7}\), and \(3 \sqrt{7}\).

Step 2 - Combine Coefficients

Add the coefficients (numbers in front of the square roots) together: 8 + 2 + 3.

Step 3 - Simplify the Coefficients

Simplify the addition: 8 + 2 + 3 = 13.

Step 4 - Rewrite the Simplified Expression

Combine the simplified coefficient with the square root: \(13 \sqrt{7}\).

Key Concepts

Identifying Like Terms

When simplifying radical expressions, our first task is to identify like terms. Like terms are terms that have the same radical part. In our example, we have the expression: \(8 \sqrt{7}+2 \sqrt{7}+3 \sqrt{7}\). All terms have the same square root of 7, making them like terms.
This is important because only like terms can be combined by their coefficients. Think of it as grouping similar items together for easy counting. For instance, you wouldn't combine apples and oranges but can easily group all apples together.

Combining Coefficients

Once we've identified the like terms in our expression, the next step is to combine the coefficients. Coefficients are the numbers directly in front of the radicals (square roots in this case). For our expression: \(8 \sqrt{7}+2 \sqrt{7}+3 \sqrt{7}\), we see that the coefficients are 8, 2, and 3.
Combining them simply involves adding these numbers together. Here is the breakdown:

• Add 8 and 2 to get 10.
• Then add 3 to 10, which results in 13.

So, 8 + 2 + 3 equals 13. Combining coefficients is similar to summing like quantities. It's like adding up all identical items in a group.

Simplified Expression

After combining the coefficients, our final task is to write the simplified expression. We've found that combining the coefficients, we get 13. Now we multiply this coefficient by the radical part that all terms share, which is \sqrt{7}\.
Thus, our simplified expression is: \(13 \sqrt{7}\).
It's important to check that the radical part remains unchanged while we adjust only the coefficients. Simplifying radical expressions can make complex problems more manageable, and this method of identifying, combining, and simplifying ensures clarity in your calculations.