Problem 16 - Elementary and Intermediate Algebra

Question

Multiply and simplify. All variables represent positive real numbers. $$ \sqrt{5} \sqrt{15} $$

Step-by-Step Solution

Answer

\( \sqrt{5} \sqrt{15} = 5\sqrt{3} \).

Step 1: Express as a Single Radical

To multiply two square roots, we first express them as a single square root by using the property \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \times b} \). So, \( \sqrt{5} \cdot \sqrt{15} = \sqrt{5 \times 15} \).

Step 2: Multiply the Numbers Inside the Radical

Perform the multiplication inside the radical. Here, we multiply 5 and 15: \( 5 \times 15 = 75 \). Thus, \( \sqrt{5 \times 15} = \sqrt{75} \).

Step 3: Simplify the Radical

To simplify \( \sqrt{75} \), we look for the largest perfect square that is a factor of 75. We find that 75 can be factored as \( 25 \times 3 \), where 25 is a perfect square. So, we express \( \sqrt{75} \) as \( \sqrt{25 \times 3} \).

Step 4: Separate and Simplify the Factors

Separate the square root of the product into the product of square roots: \( \sqrt{25 \times 3} = \sqrt{25} \cdot \sqrt{3} \). We know \( \sqrt{25} = 5 \). Therefore, \( \sqrt{75} = 5 \sqrt{3} \).

Key Concepts

Square Root Properties

Square roots behave in particular ways that make them useful for various mathematical computations.A key property of square roots is that you can multiply them together under a single radical sign. This means if you have two square roots, like \( \sqrt{a} \) and \( \sqrt{b} \), you can rewrite their product as a single square root: \( \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} \).
This property not only simplifies the process of multiplying roots but also makes it possible to easily combine and simplify expressions.

• Helpful for dealing with complex expressions where multiple radicals are involved.
• Allows direct operations inside the radical, such as multiplication.
• Simplifies further calculations.

By using these properties, you streamline the problem-solving process and can work more easily with larger numbers and complex equations.

Multiplying Radicals

Understanding how to multiply radicals is crucial since it frequently appears in math problems.When multiplying radicals, especially square roots, the process starts by using the property mentioned: a single radical sign can represent the product.
For instance, if you have \( \sqrt{5} \) and \( \sqrt{15} \), they multiply to \( \sqrt{5 \times 15} \).
Once combined under one radical, you perform the multiplication:

• Calculate the product of the numbers inside, such as \( 5 \times 15 = 75 \).
• Express the result under a single square root, resulting in expressions like \( \sqrt{75} \).

This simplification helps to see the bigger picture of the radical, making it easier to find perfect square factors and further simplify if needed.

Perfect Square Factors

Finding perfect square factors is key to simplifying radicals.A perfect square is a number that is the square of an integer, like 4, 9, 16, etc.When simplifying a radical, identify if your number can be broken down into one or more of these perfect squares.
For example, with \( \sqrt{75} \), check for the largest perfect square factor of 75, such as 25. Since 75 equals \( 25 \times 3 \), you can write:

• \( \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} \)
• Since \( \sqrt{25} = 5 \), this simplifies to \( 5\sqrt{3} \).

This process not only reduces the complexity of the expression but also allows for easier computation and comparison with other radicals.