Problem 105 - Elementary and Intermediate Algebra

Question

Look Alikes \(\cdots\) a. \(\sqrt{20}+\sqrt{20}\) b. \(\sqrt{21}+\sqrt{21}\)

Step-by-Step Solution

Answer

a. \(2\sqrt{20}\); b. \(2\sqrt{21}\)

Step 1: Understand the Problem

We need to simplify the expressions \(\sqrt{20} + \sqrt{20}\) and \(\sqrt{21} + \sqrt{21}\). The task is to combine like terms by adding the square roots.

Step 2: Simplify the Expression for Part a

For \(\sqrt{20} + \sqrt{20}\), notice that you are adding two identical terms together. In algebra, this can be simplified as \(2\sqrt{20}\), because two times something is that number times two.

Step 3: Simplify the Expression for Part b

Similarly, for \(\sqrt{21} + \sqrt{21}\), you have two identical square root terms. This can be simplified as \(2\sqrt{21}\).

Step 4: Finalize the Simplified Forms

For both expressions, the final simplified forms are: - \(\sqrt{20} + \sqrt{20} = 2\sqrt{20}\)- \(\sqrt{21} + \sqrt{21} = 2\sqrt{21}\). This is the simplest form for these expressions where the square roots cannot be simplified further to rational numbers without changing the sum of roots.

Key Concepts

Like Terms

In algebra, **like terms** are terms that have the same variable raised to the same power. They are components of algebraic expressions that we can easily combine or add together. This concept doesn’t just apply to variables; it also applies to similar numbers or expressions.
For example,

• in the expression \(3x + 2x\), both terms are like terms because they contain the variable \(x\) raised to the same power \(x^1\). Thus, they can be combined into \(5x\).
• Similarly, \sqrt{20} + \sqrt{20}\) are like terms because they are identical square root terms. This allows us to combine them to get \(2\sqrt{20}\).

Understanding like terms is crucial as it allows us to simplify expressions and makes solving equations easier. When dealing with complex algebraic expressions, recognizing and combining like terms is one of the first steps towards simplification. They must match exactly for their coefficients to be directly added or subtracted. The power here indicates that terms such as \(x^2)\ and \(x\) are not like terms as they differ in the power to which the variable \(x\) is raised.

Square Roots

A **square root** is a value that, when multiplied by itself, gives the original number. It is denoted using the symbol \(\sqrt{}\). Knowing the properties of square roots is essential for simplifying expressions involving them.
Some key properties of square roots include:

• \(\sqrt{a} \times \sqrt{a} = a\)
• \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)
• \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\)

When simplifying expressions like \(\sqrt{20} + \sqrt{20}\), it's important to recognize these as identical square roots, thus they can be simplified as \(2\sqrt{20}\). Here, you multiply the coefficient '2' by the square root \(\sqrt{20}\). When square roots are added, if they are not the same, they cannot be combined further into a single term like simple numbers and can only be simplified if they share like terms that allow combination, as shown in our examples \(\sqrt{20}+\sqrt{20}\) and \(\sqrt{21} + \sqrt{21}\).

Algebraic Expressions

An **algebraic expression** is a combination of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. These expressions are the foundation block of algebra and can range in complexity.
Key parts of algebraic expressions include:

• **Variables**: Symbols used to represent numbers, often \(x\), \(y\), or \(z\).
• **Constants**: Known values or numbers that do not change.
• **Coefficients**: Numbers multiplying the variables (e.g., in \(3x\), 3 is the coefficient).
• **Operators**: Signs like '+' and '-' that indicate the operation to be performed.

When simplifying algebraic expressions, like in the case of \(\sqrt{20} + \sqrt{20}\), we leverage the principles of like terms and arithmetic operations to make them easier to work with or solve. For more complex algebraic expressions, this simplification helps in evaluating the expressions or solving equations, making them less cumbersome to work with.