Problem 57 - Elementary and Intermediate Algebra

Question

Simplify by combining like radicals. $$ 4+\sqrt{8}+\sqrt{2}+8 $$

Step-by-Step Solution

Answer

The simplified expression is \(12 + 3\sqrt{2}\).

Step 1: Identify Like Radicals

First, identify the like radicals that can be simplified or combined. In the expression \(4+\sqrt{8}+\sqrt{2}+8\), \(\sqrt{8}\) can be simplified because it can be expressed with a simplified square root.

Step 2: Simplify Non-Prime Radicals

Simplify \(\sqrt{8}\). Recognize it as \(\sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}\). Now, rewrite the expression: \(4 + 2\sqrt{2} + \sqrt{2} + 8\).

Step 3: Combine Like Radicals

Identify the like radicals to be combined. \(2\sqrt{2}\) and \(\sqrt{2}\), add these coefficients together: \(2\sqrt{2} + \sqrt{2} = 3\sqrt{2}\).

Step 4: Combine Constant Terms

Add the constant terms in the expression: \(4 + 8 = 12\). Thus, the combined expression becomes: \(12 + 3\sqrt{2}\).

Key Concepts

Simplifying Radicals

Radicals can look complicated at first, but simplifying them helps make algebraic expressions easier to work with. A radical expression involves a root, usually a square root, but it can also involve cube roots, fourth roots, etc.

When simplifying square roots:

• Look for perfect square factors inside the radical.
• Rewrite the radical as the product of two radicals, where one of them is a perfect square.
• Simplify the perfect square radical to its simplest form.

For example, take \(\sqrt{8}\). Recognize that 8 can be broken down into \(4 \times 2\). Since \(\sqrt{4}\) is 2 (as 4 is a perfect square), the expression simplifies to \(2\sqrt{2}\). Simplifying like this allows us to later combine radicals efficiently.

Combining Like Terms

Combining like terms is critical in simplifying expressions. This means you take terms that are alike and consolidate them into one. With radicals, this process is similar to combining like terms in regular algebra.

• Identify terms with similar radical parts, such as \(\sqrt{2}\) in our example.
• Add coefficients of these like radicals together.

So, in the expression \(2\sqrt{2} + \sqrt{2}\), both terms have \(\sqrt{2}\) as the radical part.

View this as adding the coefficients (the numbers in front): 2 plus an implied 1, giving you \(3\sqrt{2}\). This consolidation simplifies calculations and offers a clearer, cleaner final expression.

Simplifying Expressions

In simplifying expressions, your goal is to reduce them to their most straightforward form, making them easier to understand or solve. The process involves a few key steps:

• Combine constant terms (like regular numbers) separately.
• Simplify radicals as discussed and combine any like radicals.
• Always check your work to ensure nothing is left in its complicated form.

For our exercise, once we've simplified \(\sqrt{8}\) and combined \(2\sqrt{2}\) and \(\sqrt{2}\) into \(3\sqrt{2}\), we add non-radical terms: 4 and 8.

This gives a simple sum of 12. Finally, the expression gets polished to \(12 + 3\sqrt{2}\), neat and efficient for any further mathematical operation.