Problem 9 - Algebra A Combined Function

Question

Use the product rule to multiply. Assume that all variables represent positive real numbers. $$ \sqrt{\frac{7}{x}} \cdot \sqrt{\frac{2}{y}} $$

Step-by-Step Solution

Answer

The result is \( \sqrt{\frac{14}{xy}} \).

Step 1: Apply the Product Rule for Square Roots

The product rule for square roots states that \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). Use this rule to combine the square roots:\( \sqrt{\frac{7}{x}} \cdot \sqrt{\frac{2}{y}} = \sqrt{\frac{7}{x} \cdot \frac{2}{y}} \).

Step 2: Simplify Inside the Square Root

Simplify the expression inside the square root by multiplying the fractions: \( \frac{7}{x} \cdot \frac{2}{y} = \frac{7 \cdot 2}{x \cdot y} = \frac{14}{xy} \).

Step 3: Write the Final Square Root Expression

Combine the results from the previous steps to write the final expression: \( \sqrt{\frac{14}{xy}} \).

Key Concepts

Understanding the Product Rule for Square Roots

When dealing with square roots, one of the essential rules is the product rule. It helps in simplifying complex square root expressions. According to this rule, the product of two square roots is the square root of the product of their radicands. If you have two numbers or expressions under square roots, you can combine them into a single square root by multiplying the expressions inside.

In algebraic terms, the rule is expressed as:

• \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \)

Applying this rule can simplify expressions, especially when dealing with fractions inside the square roots. For example, in our exercise \( \sqrt{\frac{7}{x}} \cdot \sqrt{\frac{2}{y}} \), the product rule helps us combine these into a single square root. This step reduces complexity and makes further operations, such as simplification, straightforward.

Simplifying Expressions Inside Square Roots

Once we've combined square root expressions using the product rule, the next step is simplifying what's inside. Simplification is about making the expression as straightforward as possible, often by breaking down fractions or calculating products.

• In our example, after using the product rule, we had \( \sqrt{\frac{7}{x} \cdot \frac{2}{y}} \).
• We then multiply the fractions: \( \frac{7}{x} \cdot \frac{2}{y} \) to get \( \frac{7 \cdot 2}{x \cdot y} = \frac{14}{xy} \).

By multiplying the numerators and denominators separately, and when possible, reducing the fraction further, we make the expression within the square root more manageable. This step is crucial for accurate problem-solving and plays a significant role in reaching the final simplified expression.

The Role of Multiplying Fractions in Simplification

Multiplying fractions is another key process in algebra, essential for solving problems that involve expressions with fractions. It involves straightforward steps that, rightfully applied, lead to simplification.

Here's a refresher on how to multiply fractions:

• Multiply the numerators together: If you have \( \frac{a}{b} \) and \( \frac{c}{d} \), the product of the numerators is \( a \cdot c \).
• Multiply the denominators together: So for the above example, the product of the denominators is \( b \cdot d \).
• Write the result as a new fraction: The result is \( \frac{a \cdot c}{b \cdot d} \).

Applying this method helps to simplify the inside of a square root or any other expression involving fractions. In our example, we went from \( \sqrt{\frac{7}{x}} \cdot \sqrt{\frac{2}{y}} \) to the simpler form \( \sqrt{\frac{14}{xy}} \). This transformation is pivotal for understanding how various algebraic expressions interact and simplify within complex equations.