Problem 36 - College Algebra

Question

Simplify each expression. $$\sqrt{4 y^{2}}$$

Step-by-Step Solution

Answer

\(2y\)

Step 1: Identify the Expression Inside the Radical

We start with the expression inside the square root, which is \(4y^2\). Our goal is to simplify this expression.

Step 2: Simplify the Expression Inside the Radical

The expression \(4y^2\) consists of the coefficient \(4\) and the variable raised to a power \(y^2\). Both of these can be considered perfect squares, as \(4 = 2^2\) and \(y^2 = (y)^2\).

Step 3: Apply the Square Root to Each Component

We apply the square root to each component separately:\[ \sqrt{4y^2} = \sqrt{4} \times \sqrt{y^2} \]This simplifies to:\[ 2 \times y \]

Step 4: Combine the Results

The final step is to combine the results of the square roots: \[ \sqrt{4y^2} = 2y \]Thus, the simplified expression is \(2y\).

Key Concepts

Perfect Squares

A perfect square is a number or algebraic expression that is the square of an integer or another algebraic expression. Understanding perfect squares is crucial for simplifying radical expressions. For instance, if you have the number 4, it is a perfect square because it can be written as the square of 2, i.e., \(4 = 2^2\). Similarly, an algebraic term like \(y^2\) is a perfect square because it is the square of the expression \((y)\).

Recognizing perfect squares is important when simplifying radical expressions because it allows you to easily "remove" the square root by expressing the number or algebraic term as a square and then simplifying it. For example, when you see \(4y^2\), you can think of it as \((2)^2 (y)^2\), which makes the simplification process straightforward. The key is spotting these perfect squares so you can break them down using their roots, drastically simplifying your task.

Square Root

The square root is a mathematical operation that determines what number, when multiplied by itself, equals the original number. This concept is often represented with the radical symbol \(\sqrt{}\). For example, the square root of 4, written as \(\sqrt{4}\), is 2, because \(2 \times 2 = 4\).

When dealing with algebraic expressions, the square root functions in a similar fashion. Taking the square root of \(y^2\), for instance, results in \(y\) because \(y \times y = y^2\).

Square roots play a pivotal role in simplifying radical expressions. By finding the square root of perfect squares within the expression, you can simplify complex radicals into more manageable algebraic terms. It’s all about looking for components that are perfect squares and taking their root to simplify the entire expression.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. Simplifying these expressions involves reducing them to their simplest form without changing their value. This often includes factoring, expanding, or applying operations like addition or multiplication.

When you simplify an expression that includes radical operations, like \(\sqrt{4y^2}\), your main goal is to reduce the complexity by removing the radicals when possible. This usually involves finding and simplifying any perfect squares in the expression. For instance, in \(4y^2\), both 4 and \(y^2\) are perfect squares.

• First, identify the perfect squares within the expression.
• Next, apply the square root to these perfect squares individually.
• Finally, multiply the results together to get a simplified algebraic expression.

The final simplified form maintains the original value but is more concise and easier to use in further mathematical operations.