Problem 1 - Algebra 2

Question

Add or subtract if possible. $$ 5 \sqrt{6}+\sqrt{6} $$

Step-by-Step Solution

Answer

The answer to the problem is \(6 \sqrt{6}\).

Step 1 Identify like radicals

Identify like radicals in the equation. Here, our like radicals are \(5 \sqrt{6}\) and \(\sqrt{6}\)

Step 2 Add the radicals

Add the coefficients. We add the coefficients of these like radicals like normal numbers: 5 (from \(5 \sqrt{6}\)) + 1 (from \(\sqrt{6}\), you assume there's 1 when no coefficient is written in front of the radical) = 6.

Step 3 Write the final result

We place the resultant coefficient in front of our like radical to get the result. Thus, the answer is \(6 \sqrt{6}\).

Key Concepts

Like Radicals

When dealing with radicals, one of the most important concepts is identifying like radicals. Like radicals are similar to like terms in algebra, meaning they must have the same radicand (the number under the radical) and the same index (the degree of the root). For example, in the expression \(5\sqrt{6} + \sqrt{6}\), both involve the square root of 6, making them like radicals. It is crucial to note that only like radicals can be combined through addition or subtraction, just as only like terms can be combined in algebraic expressions. By recognizing like radicals, we can simplify expressions efficiently and ensure that calculations are accurate.

Addition of Radicals

Adding radicals can often be simplified by first identifying and combining like radicals. In the case of adding \(5\sqrt{6}\) and \(\sqrt{6}\), we note that they are like radicals because they both involve \(\sqrt{6}\). To add them, we focus on their coefficients, the numbers in front of the radicals. The first term, \(5\sqrt{6}\), has a coefficient of 5, and the second term, \(\sqrt{6}\), has an implicit coefficient of 1 (since it's not explicitly written, we assume it to be 1). Here's how to add them:

• Add the coefficients: 5 + 1 = 6
• Keep the radical part the same, because we are only combining the coefficients.

Thus, the result of adding these radicals is \(6\sqrt{6}\). By mastering the addition of radicals, we become more adept at organizing and simplifying algebraic expressions.

Algebraic Expressions

Algebraic expressions often involve combinations of numbers, variables, and operational symbols, including radicals. Simplifying such expressions requires understanding concepts like like radicals and addition of radicals, among others. When working with algebraic expressions that include radicals, it is crucial to:

• Identify similar components such as like terms or radicals to simplify effectively.
• Apply rules of arithmetic correctly to handle coefficients and constants.
• Watch for opportunities to simplify roots and expressions before combining them, particularly if the terms change via multiplication or division.

For instance, in the expression \(5\sqrt{6} + \sqrt{6}\), identifying the like radicals allows us to quickly combine them by adding their coefficients. Simplifying algebraic expressions requires deep understanding and careful application of algebraic principles to ensure that all calculations are accurate and efficient.