Problem 20 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \frac{\sqrt{4 x y^{2}}}{\sqrt{x}} $$

Step-by-Step Solution

Answer

The simplified expression is \( 2y \).

Step 1: Simplify the Radicals

Simplify both the numerator and the denominator of the expression. The expression is \( \frac{\sqrt{4xy^{2}}}{\sqrt{x}} \). The numerator, \( \sqrt{4xy^{2}} \), can be broken down as \( \sqrt{4} \cdot \sqrt{x} \cdot \sqrt{y^{2}} \).Since \( \sqrt{4} = 2 \) and \( \sqrt{y^{2}} = y \), the numerator becomes \( 2y\sqrt{x} \). The denominator, \( \sqrt{x} \), stays the same.

Step 2: Divide the Radicals

Now that we have simplified the numerator, divide it by the denominator. The expression becomes:\[ \frac{2y \sqrt{x}}{\sqrt{x}} \]Since both the numerator and the denominator include \( \sqrt{x} \), they cancel each other out, simplifying to \( 2y \).

Step 3: Final Simplification

After canceling \( \sqrt{x} \) in both the numerator and the denominator, we are left with the expression \( 2y \). Thus, the expression \( \frac{\sqrt{4xy^{2}}}{\sqrt{x}} \) simplifies to \( 2y \).

Key Concepts

Radicals

Radicals, often symbolized by the square root sign, are essential components in algebra. You will frequently encounter them when simplifying expressions involving roots. Understanding how to simplify radicals is crucial.

Radicals represent the root of a number. The most common is the square root, but it can denote other roots like cube roots. The expression inside the radical symbol is called the radicand.

• To simplify a radical, break it down into its prime factors if possible. This makes it easier to identify perfect squares or cubes, which can be simplified further.
• A perfect square is a number that can be expressed as the square of an integer, such as 4, 9, or 16. For example, \(\sqrt{4} = 2\).
• This process of breaking down is called simplification, which makes working with radicals much easier in equations.

When simplifying radicals with variables, remember the rules of exponents. For instance, \(\sqrt{y^2} = y\) because it is a perfect square. In the given exercise, simplify each component separately to manage complex radical expressions effectively.

Algebraic Expressions

An algebraic expression is a combination of variables, numbers, and arithmetic operations. Radicals often integrate into these expressions, offering a more complex layer of algebra.

When simplifying algebraic expressions with radicals, the goal is to reduce them to their simplest form while maintaining their value. Here are some key points to consider:

• Identify like terms. In this exercise, both the numerator and the denominator have the radical \(\sqrt{x}\) which can simplify the expression when divided.
• Factor out common terms. This helps in rewriting the expression in a simpler form.
• The simplification of radicals in algebraic expressions often involves breaking down constants and variables inside the radical, as shown in the exercise where \(\sqrt{4xy^2} = 2y\sqrt{x}\).

By identifying similar radicals or understanding variable powers, you can simplify an algebraic expression efficiently.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating them. It allows expression and manipulation of mathematical relationships.

When approaching problems like the one provided, remember that algebra is about finding the simplest form of expressions while maintaining equality.

• Use operations strategically to simplify expressions. Recognize opportunities to cancel or factor out terms, as seen when simplifying \(\frac{\sqrt{4xy^2}}{\sqrt{x}}\).
• Algebra provides the tools to balance equations. By simplifying terms, we can solve equations more easily.
• Radicals are just another component of algebraic expressions, necessitating methods such as simplification or cancellation to streamline equations.

The exercise taught us how radicals interact within larger expressions and showcased algebra's power in simplifying these constructs to more manageable forms. Understanding the roots, breaking down each component, and strategically canceling common elements lead to a cleaner, more straightforward solution.