Problem 27 - Algebra A Combined Function

Question

Use the quotient rule to simplify. Assume that all variables represent positive real numbers. $$ \sqrt{\frac{5 x^{2}}{4 y^{2}}} $$

Step-by-Step Solution

Answer

\( \frac{x\sqrt{5}}{2y} \)

Step 1: Understand the Expression

The given expression is \( \sqrt{\frac{5x^2}{4y^2}} \). This is a square root of a fraction.

Step 2: Apply the Quotient Rule for Square Roots

The square root of a fraction \( \sqrt{\frac{a}{b}} \) can be expressed as \( \frac{\sqrt{a}}{\sqrt{b}} \). Apply this rule to the given expression. We have:\[ \sqrt{\frac{5x^2}{4y^2}} = \frac{\sqrt{5x^2}}{\sqrt{4y^2}} \]

Step 3: Simplify the Numerator

The numerator is \( \sqrt{5x^2} \). Express it as \( \sqrt{5} \cdot \sqrt{x^2} \). Since \( \sqrt{x^2} = x \), we have \[ \sqrt{5x^2} = \sqrt{5} \cdot x \]

Step 4: Simplify the Denominator

The denominator is \( \sqrt{4y^2} \). Express it as \( \sqrt{4} \cdot \sqrt{y^2} \). Since \( \sqrt{4} = 2 \) and \( \sqrt{y^2} = y \), we have \[ \sqrt{4y^2} = 2y \]

Step 5: Form the Simplified Expression

Substitute the simplified numerator and denominator back into the expression:\[ \frac{\sqrt{5x^2}}{\sqrt{4y^2}} = \frac{\sqrt{5}\cdot x}{2y} \]

Step 6: Final Answer

The simplified expression is:\[ \frac{x\sqrt{5}}{2y} \]

Key Concepts

Square Root of a Fraction

Understanding the concept of a square root of a fraction can simplify many algebraic expressions. When you encounter \( \sqrt{\frac{a}{b}} \), you can separate it into two parts: the square root of the numerator and the square root of the denominator.

• This means: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \).

This rule allows you to handle the top and bottom separately, making the simplification process easier.
For example, when applying this to \( \sqrt{\frac{5x^2}{4y^2}} \), you can rewrite it as:

• Numerator: \( \sqrt{5x^2} \)
• Denominator: \( \sqrt{4y^2} \)

This step-by-step dissection helps make complex-looking expressions much more manageable.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them to a form that is easier to understand or work with.
The goal is to make the expression less complicated while retaining its original value. In the given exercise,
we apply the quotient rule, which is particularly useful for expressions involving fractions and square roots.
Start by breaking down each component. For the numerator \( \sqrt{5x^2} \), you separate it into \( \sqrt{5} \cdot \sqrt{x^2} \).

• Remember: \( \sqrt{x^2} = x \).

For the denominator \( \sqrt{4y^2} \), use the fact that \( \sqrt{4} = 2 \) and \( \sqrt{y^2} = y \).
Substituting these back gives:

• \( \frac{x \cdot \sqrt{5}}{2y} \)

This process highlights how simplifying each part of the expression individually leads to the overall simplified expression.

Variables in Algebra

Variables are symbols used to represent numbers in algebra.
They are crucial for forming expressions, equations, and understanding relationships in mathematics.
In the given exercise, variables \( x \) and \( y \) illustrate how algebraic expressions can include these entities to signify unknowns.
When working with variables:

• Keep in mind the operations you apply also affect them just like numeric values.
• Variables follow the same mathematical rules, such as exponent rules and arithmetic operations.

Simplifying expressions with variables requires you to respect these rules while applying algebraic techniques.
This ensures that the expression is accurate and correctly represents the intended relationship.

Properties of Exponents

Understanding the properties of exponents is essential when simplifying algebraic expressions, especially those involving square roots.
Here are a few key points to keep in mind:

• \( x^2 \) means \( x \cdot x \).
• Taking a square root, as in \( \sqrt{x^2} \), essentially means reverting \( x \cdot x \) back to \( x \).

These properties come in handy when dealing with expressions such as \( \sqrt{5x^2} \) and \( \sqrt{4y^2} \).
Knowing that \( \sqrt{x^2} = x \) allows us to simplify these expressions efficiently by removing the square, leaving the base variable.
Applying the right properties ensures the expression is simplified correctly, adhering to the mathematical principles of exponents.