Problem 36 - College Algebra

Question

For the following exercises, simplify each expression. $$ \sqrt{4 y^{2}} $$

Step-by-Step Solution

Answer

\( 2|y| \)

Step 1: Understand the Square Root

When dealing with a square root, you are finding the value that, when multiplied by itself, gives the number under the root. For \( \sqrt{4y^2} \), you want to find a number that, when squared, results in \(4y^2\).

Step 2: Simplify the Square of a Constant and a Variable

Separate the square root into two parts: the constant and the variable expression.\( \sqrt{4y^2} = \sqrt{4} \times \sqrt{y^2} \).

Step 3: Solve Square Root of the Constant

Calculate \(\sqrt{4}\). Since 2 squared equals 4, \(\sqrt{4} = 2\).

Step 4: Simplify the Variable Expression

Find \(\sqrt{y^2}\). By definition, the square root of a square (\(y^2\)) is the absolute value of the base, so \(\sqrt{y^2} = |y|\).

Step 5: Combine the Results

Multiply the results from Step 3 and Step 4 to get the simplified expression.Thus, \( \sqrt{4y^2} = 2\cdot |y| = 2|y| \).

Key Concepts

Square Roots

Square roots are a foundational concept in algebra, representing the inverse of squaring a number. When you take the square root of a number, you're determining what number, when multiplied by itself, equates to the original number under the root sign.
For example, the square root operation is denoted by the radical symbol \(\sqrt{}\). When you see \(\sqrt{4}\), you're finding the number that, when squared, will produce 4. In this case, it is 2, since \(2 \times 2 = 4\).

• It is important to remember that numbers have two square roots: a positive and a negative. However, the square root symbol \(\sqrt{}\) typically denotes the principal (positive) square root.
• When simplifying square roots that involve products, like \(\sqrt{4y^2}\), you can use the property \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\).

Understanding how square roots work is essential for further explorations in algebra.

Variable Expressions

Variable expressions include any combination of numbers, variables, and operations like addition, subtraction, multiplication, and division. Variable expressions often appear in algebra problems, representing quantities that can change.
Variables are typically represented by letters (like \(y\) in our example). When you see a term such as \(y^2\), this denotes that \(y\) is squared. In variable expressions, both constant terms and variable parts can interact together in various algebraic manipulations.

• Variables can represent unknown values, and solving variable expressions often involves simplifying them or solving for the variable.
• In simplification tasks, checking the power of variables is crucial. A variable raised to a power, such as \(y^2\), follows exponent rules when simplifying.

Working with variable expressions allows for modeling numerous real-world problems in mathematical terms.

Absolute Value

Absolute value describes the distance of a number from zero on the number line, irrespective of its sign. It is denoted by two vertical bars around the number or expression, such as \(|y|\). Absolute value converts any negative number into a positive one.
When simplifying expressions with square roots like \(\sqrt{y^2}\), the result is \(|y|\). This accounts for the fact that while \(y^2\) is always non-negative, \(y\) itself could be negative or positive.

• The expression \(|y|\) ensures results remain non-negative which aligns with the principal property of square roots.
• Understanding absolute value is key in contexts where magnitude matters more than direction, such as measuring distances or quantities.

The concept ensures that calculations involving square roots consider both positive and negative roots.

Constants in Algebra

Constants are fixed values that do not change. In algebra, constants can serve as known quantities within an expression or equation. For example, in the expression \(\sqrt{4y^2}\), the number 4 is a constant.
Simplifying algebraic expressions often involves dealing with these constants first. For instance, calculating \(\sqrt{4}\) is straightforward since you know it equals 2.

• Constants provide a stable reference point when performing algebraic operations.
• They are crucial in understanding how expressions collapse into simpler forms, as they guide the initial steps in simplification.

Recognizing and working confidently with constants is critical for mastering algebraic manipulations.