Problem 7 - Intermediate Algebra

Question

Fill in the blanks. Consider the expressions \(\sqrt{4 \cdot 5}\) and \(\sqrt{4} \sqrt{5} .\) Which expression is a. the square root of a product? b. the product of square roots? c. How are these two expressions related?

Step-by-Step Solution

Answer

a. \( \sqrt{4 \cdot 5} \) is the square root of a product. b. \( \sqrt{4} \sqrt{5} \) is the product of square roots. c. Both expressions are equivalent by the square root property.

Step 1: Identify the Expression Components

Identify the components of the two expressions:Expression 1 is \( \sqrt{4 \cdot 5} \), which involves taking the square root of the product of 4 and 5.Expression 2 is \( \sqrt{4} \sqrt{5} \), which involves the product of the individual square roots of 4 and 5.

Step 2: Determining the Square Root of a Product

The phrase "square root of a product" refers to taking the square root of the product of two or more numbers. In this case: Expression 1: \( \sqrt{4 \cdot 5} \) is the square root of the product of 4 and 5.

Step 3: Determining the Product of Square Roots

The phrase "product of square roots" means multiplying the square roots of individual numbers. In this case:Expression 2: \( \sqrt{4} \sqrt{5} \) is the product of the square roots of 4 and 5.

Step 4: Relationship Between Expressions

To determine the relationship between these expressions, consider the property of square roots that allows us to break up or combine products, known as the square root of a product property \( \sqrt{a \cdot b} = \sqrt{a} \times \sqrt{b} \):The expressions \( \sqrt{4 \cdot 5} \) and \( \sqrt{4} \times \sqrt{5} \) are equivalent based on this property.

Key Concepts

Product of Square Roots

This refers to multiplying the square roots of individual numbers together. It involves working with separate square root terms, each originating from different numbers. For example, consider two numbers, 4 and 5. When you calculate the square root for each one and then multiply them, this forms the product of square roots. So the expression

• \( \sqrt{4} \)
• \( \sqrt{5} \)

When you multiply these, you get the expression \( \sqrt{4} \times \sqrt{5} \). This operation emphasizes that each square root stands alone before they interact through multiplication.

Using a specific case, compute:

• \( \sqrt{4} = 2 \)
• \( \sqrt{5} \approx 2.236 \)

Multiplying these results gives approximately 4.472, showing clearly the idea behind the product of square roots.

Square Root of a Product

This concept takes a different approach by addressing how a product of numbers can be simplified directly into a single square root. Starting with two numbers, for example, let's say they are 4 and 5, you can first multiply these numbers together, as in 4 \( \times \) 5 = 20. Then, you apply the square root to this product.

Hence, when you write \( \sqrt{4 \cdot 5} \), you are diving into the square root of the product. This expression simplifies directly to \( \sqrt{20} \). The simplicity and compactness of writing the product within a single square root is the main feature of this concept.

The method is useful because:

• Simplification is performed in one step.
• Helps in working with larger numbers, as breaking them into smaller multiplicands often eases calculation.
• Demonstrates the flexibility and power of dealing with roots.

Equivalent Expressions

Although at first glance, \( \sqrt{4 \cdot 5} \) and \( \sqrt{4} \times \sqrt{5} \) may seem different, a special mathematical property links them. This is known as the square root of a product property, which states:\[ \sqrt{a \cdot b} = \sqrt{a} \times \sqrt{b}\]
The beauty here is realizing that any product of two numbers underneath a square root can be split into individual roots and vice versa.

Consider our earlier example:

• \( \sqrt{4 \cdot 5} = \sqrt{20} \)
• \( \sqrt{4} \times \sqrt{5} \approx 4.472 \)

Both resolve to approximately the same numeric value, revealing them as equivalent expressions. This equivalence provides flexibility in solving problems, either simplifying a single square root or simplifying through multiplication of separate square roots. This principle can save time and effort in many mathematical tasks!