Problem 29 - Intermediate Algebra

Question

Multiply and simplify. All variables represent positive real numbers. $$ \sqrt{2}(4 \sqrt{6}+2 \sqrt{7}) $$

Step-by-Step Solution

Answer

\(8\sqrt{3} + 2\sqrt{14} \)

Step 1: Distribute the Square Root

Distribute \( \sqrt{2} \) to both terms inside the parentheses. This gives \( \sqrt{2} \times 4\sqrt{6} + \sqrt{2} \times 2\sqrt{7} \).

Step 2: Simplify Each Term

Simplify \( \sqrt{2} \times 4\sqrt{6} \) by using the property \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \). We get \( 4\sqrt{2 \times 6} = 4\sqrt{12} \). For the second term, \( \sqrt{2} \times 2\sqrt{7} = 2\sqrt{2 \times 7} = 2\sqrt{14} \).

Step 3: Simplify the Square Roots

Simplify \( \sqrt{12} \) into \( \sqrt{4 \times 3} = 2\sqrt{3} \). So, \( 4\sqrt{12} = 4(2\sqrt{3}) = 8\sqrt{3} \). \( \sqrt{14} \) is already in its simplest form. Thus the terms simplify to \( 8\sqrt{3} + 2\sqrt{14} \).

Step 4: Combine the Results

Combine the simplified terms for the final expression. The resulting simplified expression is \( 8\sqrt{3} + 2\sqrt{14} \).

Key Concepts

Understanding the Distributive Property with Radicals

The Distributive Property is a foundational concept in algebra that helps in simplifying expressions, especially when dealing with radicals. It involves distributing a single factor across terms within parentheses. In the context of radical expressions, it guides us to distribute the square root across each term in the parentheses separately.

Let's consider this task as an example: \( \sqrt{2}(4 \sqrt{6} + 2 \sqrt{7}) \). We use the distributive property to deal with the terms inside the parentheses. This process involves multiplying \( \sqrt{2} \) by each term inside separately, resulting in two products: \( (\sqrt{2} \times 4 \sqrt{6}) + (\sqrt{2} \times 2 \sqrt{7}) \).

By following the distributive property correctly, you lay the groundwork for simplifying each resulting term further. This is a crucial step for achieving the final simplified expression while handling complex equations with radicals.

Simplifying Square Roots in Radical Expressions

Simplifying square roots is an essential skill when solving problems with radicals. The goal is to simplify the square root into the lowest possible form, making final calculations more straightforward.

To simplify square roots, we identify any perfect squares within the radicand (the number inside the square root). For example, when simplifying \( \sqrt{12} \), we notice that 12 can be rewritten as \( 4 \times 3 \). Since \( 4 \) is a perfect square and \( \sqrt{4} = 2 \), we simplify: \( \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \).

• Look for factors that are perfect squares within the radicand.
• Simplify by extracting these perfect square factors out of the square root.

In our exercise, while \( \sqrt{14} \) cannot be simplified further as none of its factors are perfect squares, \( \sqrt{12} \) reduces beautifully to \( 2\sqrt{3} \).

Mastering this simplification process is key to reducing expressions efficiently and effectively.

Multiplying Radicals Simplified

Multiplying radicals involves using a specific property of square roots: \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \). This property allows us to combine or break apart square roots during multiplication, making it easier to manage radical expressions.

Consider multiplying \( \sqrt{2} \times 4\sqrt{6} \). This breaks down as follows:

• Combine the radicals: \( \sqrt{2} \times \sqrt{6} = \sqrt{12} \).
• Don't forget to multiply outside numbers as well: \( 4 \times \sqrt{12} \) results in \( 4\sqrt{12} \).

Similarly, for the term \( \sqrt{2} \times 2\sqrt{7} \), multiply the radicals together: \( \sqrt{2 \times 7} = \sqrt{14} \) and the numbers outside the radicals: \( 2\sqrt{14} \).

By understanding and applying the rule for multiplying radicals, you're able to transition from separate square roots into more manageable terms, thus simplifying your work with complex expressions.