Problem 53 - Intermediate Algebra

Question

Simplify by combining like radicals. $$ 5 \sqrt{7}+3 \sqrt{7} $$

Step-by-Step Solution

Answer

The simplified expression is \(8 \sqrt{7}\).

Step 1: Identify Like Radicals

Examine the expression to identify like radicals. Both terms, \(5 \sqrt{7}\) and \(3 \sqrt{7}\), have the same radical part under the square root, which is \(\sqrt{7}\). This makes them like radicals.

Step 2: Combine the Coefficients

Since the radicals are like terms, combine them by adding their coefficients: \(5 + 3\). This gives us \(8\).

Step 3: Construct the Simplified Expression

Write the simplified form by multiplying the combined coefficient by the radical part: \(8 \sqrt{7}\).

Key Concepts

Simplifying Radicals

When you come across problems that ask you to simplify radicals, focus first on the numbers under the square root, known as radicals. Identifying these correctly is crucial. Radicals become 'like radicals' when the number under the square root is the same in each term involved. For example, in the expression \(5 \sqrt{7} + 3 \sqrt{7}\), both terms involve the same radical part, \(\sqrt{7}\). This similarity is key, because only like radicals can be combined directly; much like how you combine like terms in algebra. But before performing any arithmetic on the coefficients, you must ensure that the radicals are indeed identical. If they are different, further simplification is not possible unless the expression can be rewritten to make them identical.

Combining Like Terms

The concept of combining like terms is widely used in algebra. It's a way to simplify mathematical expressions and make them easier to work with.In the context of radicals, once you've identified like radicals, their coefficients can be combined just as you would with simple numbers. This means adding or subtracting them based on the operation involved. In the exercise \(5 \sqrt{7} + 3 \sqrt{7}\), after confirming that both terms are like radicals, you simply need to add their coefficients:

• Add 5 and 3, which gives you 8.
• Write the result with the common radical part: \(8 \sqrt{7}\).

Combining terms in this way can greatly simplify complex expressions, turning them into easier forms to manipulate in further calculations.

Coefficients

The coefficient plays a pivotal role in expressions, including those involving radicals. A coefficient is the number placed in front of a variable or radical.Consider the original expression \(5 \sqrt{7} + 3 \sqrt{7}\). Here, 5 and 3 are the coefficients. They indicate how many times the radical part, or \(\sqrt{7}\), is counted. Understanding coefficients is important because they determine the scaling of the terms you combine. When combining \(5 \sqrt{7}\) and \(3 \sqrt{7}\), you are essentially finding a total based on these counts, turning them into \(8 \sqrt{7}\) with the new coefficient being the sum of the original coefficients. It's always a good practice to focus on coefficients when simplifying, as they impact the overall magnitude of the expression. By mastering this concept, you make it easier to work with more complex mathematical problems.