Problem 62 - Algebra 2

Question

Simplify each expression. $$ 5 \sqrt{5}+\sqrt{5} $$

Step-by-Step Solution

Answer

Simplified expression is \(6\sqrt{5}\).

Step 1: Identify Like Terms

Like terms are terms that contain the same variables raised to the same power. In the expression \(5\sqrt{5} + \sqrt{5}\), there are two like terms that can be combined, which are \(5\sqrt{5}\) and \(\sqrt{5}\).

Step 2: Combine Like Terms

When combining like terms, you add or subtract the coefficients. In this case, \(\sqrt{5}\) can be re-written as \(1\sqrt{5}\). So, adding the coefficients of \(5\sqrt{5}\) and \(1\sqrt{5}\) would result in \(5 + 1\) times \(\sqrt{5}\) which equals to \(6\sqrt{5}\).

Key Concepts

Like Terms

When working with expressions, it's essential to understand the concept of like terms. Like terms are simply terms that have the same variables raised to the same power. This means they can be grouped together when simplifying an expression.
For example, in the expression \(5\sqrt{5} + \sqrt{5}\), both terms are like terms because they both contain the same radical, \(\sqrt{5}\). The coefficients (numbers in front of the terms) may differ, but since the radicals are identical, these can be combined.
Recognizing like terms is crucial in simplifying expressions, as it allows for advanced calculations. It's all about identifying similarities and understanding that parts of an expression that look alike can often be combined to make solving easier.

Radical Expressions

Radical expressions are mathematical expressions that contain a radical symbol, \(\sqrt{}\), which is used to denote the root of a number. In our example, \(\sqrt{5}\) represents the square root of 5.

• Radicals can include square roots, cube roots, or any higher-order roots.
• The number under the root, known as the radicand, determines what you're taking the root of.

Simplifying radical expressions often involves operations with these radicals, such as combining or factoring them. Keep in mind that radicals are only considered like terms if they have the same radicand.
Understanding radical expressions allows you to manipulate them efficiently in calculations and helps in various mathematical fields, including algebra and calculus.

Combining Coefficients

Combining coefficients is a fundamental concept in simplifying expressions. Coefficients are the numerical factors that multiply the variables or radicals in an expression.
In our example, \(5\sqrt{5} + \sqrt{5}\), notice that the coefficients are 5 and 1 (the \(1\) is implied in \(\sqrt{5}\) because it's equivalent to \(1\sqrt{5}\)).
To combine coefficients, we simply add the numbers in front of like terms:

• Start by ensuring the terms you're combining have identical radicals or variables.
• Add the coefficients of the like terms while keeping the variable or radical part constant.

So here, by adding the coefficients 5 and 1, we get 6, resulting in the final simplified form of \(6\sqrt{5}\).
This method makes processing and simplifying algebraic expressions or equations more manageable by reducing the complexity of the expression.