Problem 12 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \sqrt{\frac{4}{y}} \cdot \sqrt{\frac{y}{5}} $$

Step-by-Step Solution

Answer

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

Step 1: Simplify the Product of Square Roots

Start by using the property of square roots that states \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). In this case, we combine the two square roots: \[ \sqrt{\frac{4}{y}} \cdot \sqrt{\frac{y}{5}} = \sqrt{\frac{4}{y} \cdot \frac{y}{5}}. \]

Step 2: Multiply the Fractions Under the Square Root

Next, simplify the expression under the square root by multiplying the fractions. To do this, multiply the numerators and multiply the denominators: \[ \frac{4}{y} \cdot \frac{y}{5} = \frac{4 \cdot y}{y \cdot 5}. \]

Step 3: Cancel Common Factors

Notice that \( y \) appears in both the numerator and the denominator. Cancel \( y \) from both to simplify: \[ \frac{4 \cdot y}{y \cdot 5} = \frac{4}{5}. \]

Step 4: Evaluate the Square Root

Now, take the square root of the result from Step 3: \[ \sqrt{\frac{4}{5}}. \] The expression simplifies further because \( \sqrt{4} = 2 \), resulting in: \[ \frac{2}{\sqrt{5}}. \]

Step 5: Rationalize the Denominator

To rationalize the denominator, multiply the numerator and denominator by \( \sqrt{5} \): \[ \frac{2}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{2 \sqrt{5}}{5}. \]

Key Concepts

square roots

Square roots are used to determine what number, when multiplied by itself, results in a given number. In mathematical notation, the square root of a number is represented as \( \sqrt{a} \), where \( a \) is the number you want to find the square root of.
Here is what you should know about square roots:

• A square root is only for non-negative numbers if dealing with real numbers. Complex numbers, however, can have negative square roots.
• When multiplying square roots together, you can combine them under a single square root. For example, \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \).
• Square roots of perfect squares (like 4, 9, 16) are integers. For example, \( \sqrt{4} = 2 \) and \( \sqrt{9} = 3 \).

Understanding the properties and operations involving square roots is crucial, especially when simplifying expressions as in the given problem. It allows for the consolidation of multiple square roots into a simpler, single square root expression.

rationalizing denominators

Rationalizing the denominator refers to the process of eliminating any square roots or irrational numbers from the bottom (denominator) of a fraction. This is typically done to simplify the expression and make it easier to work with or further calculate.
Here is a simple guide to rationalize denominators:

• When you have a square root in the denominator, you can multiply both the top (numerator) and bottom (denominator) of the fraction by that same square root to eliminate it.
• For example, if you have \( \frac{2}{\sqrt{5}} \), multiply both numerator and denominator by \( \sqrt{5} \) to get \( \frac{2 \sqrt{5}}{5} \). This process removes the square root from the denominator.
• The goal is to have a rational number left in the denominator.

This method makes further calculations cleaner and is often required in many mathematical contexts.

multiplying fractions

Multiplying fractions might initially seem complex, but it's straightforward once you understand the procedure. When you multiply fractions, you deal directly with the numerators and denominators.
Here's how to multiply fractions:

• Multiply the numerators of the fractions together to get the new numerator.
• Multiply the denominators of the fractions together to get the new denominator.
• Simplify the resulting fraction if possible by finding common factors.

For example, consider the fractions \( \frac{4}{y} \) and \( \frac{y}{5} \). When multiplied, the result is \( \frac{4 \cdot y}{y \cdot 5} \). Notice that if there are common factors in the numerator and denominator, like \( y \) here, they can be canceled, simplifying the expression to \( \frac{4}{5} \). Understanding this fundamental operation is key in simplifying algebraic expressions, particularly when combined with operations like finding square roots and rationalizing denominators.