Problem 11 - Intermediate Algebra

Question

Perform each operation, if possible. a. \(4 \sqrt{6}+2 \sqrt{6}\) b. \(4 \sqrt{6}(2 \sqrt{6})\) c. \(3 \sqrt{2}-2 \sqrt{3}\) d. \(3 \sqrt{2}(-2 \sqrt{3})\)

Step-by-Step Solution

Answer

a. \(6\sqrt{6}\); b. \(48\); c. \(3\sqrt{2} - 2\sqrt{3}\); d. \(-6\sqrt{6}\).

Step 1: Identify like terms (a)

For part (a), we notice that both terms, \(4 \sqrt{6}\) and \(2 \sqrt{6}\), have the same radical part, \(\sqrt{6}\). Therefore, they are like terms and can be combined by simply adding their coefficients: \[4\sqrt{6} + 2\sqrt{6} = (4+2)\sqrt{6} = 6\sqrt{6}.\]

Step 2: Multiply radicals (b)

In part (b), we multiply \(4 \sqrt{6}\) by \(2 \sqrt{6}\). First, multiply the coefficients: \[4 \times 2 = 8.\]Then, multiply the radicals:\[\sqrt{6} \times \sqrt{6} = \sqrt{6^2} = \sqrt{36} = 6.\]Combine the results: \[8 \times 6 = 48.\]

Step 3: Evaluate different radicals (c)

In part (c), the terms \(3 \sqrt{2}\) and \(2 \sqrt{3}\) involve different radicals, \(\sqrt{2}\) and \(\sqrt{3}\). Since these are not like terms, we cannot combine them any further. Therefore, the expression remains as it is:\[3\sqrt{2} - 2\sqrt{3}.\]

Step 4: Multiply with different radicals (d)

In part (d), multiply \(3 \sqrt{2}\) by \(-2 \sqrt{3}\). Multiply the coefficients first:\[3 \times (-2) = -6.\]Then, multiply the radicals: \[\sqrt{2} \times \sqrt{3} = \sqrt{6}.\]Combine the results:\[-6 \times \sqrt{6} = -6 \sqrt{6}.\]

Key Concepts

Understanding Like Terms in Radicals

Radical expressions can have what we call "like terms." Like terms are terms that have the same radical part. This means their square roots are the same. When terms are "like," we can combine them much like regular algebraic expressions. Let’s break it down:

• If you have two or more radical terms like \(4 \sqrt{6}\) and \(2 \sqrt{6}\), they can be combined by adding their coefficients together since the \(\sqrt{6}\) part is identical for both terms.
• In our example, to combine them: add the coefficients (4 and 2) together, and keep the like radical part the same. This gives us \( (4+2) \sqrt{6} = 6 \sqrt{6}. \)
• If the radical parts are different, such as \(\sqrt{2}\) and \(\sqrt{3}\), we cannot combine them in the same way, as they are not like terms.

Identifying like terms in radical expressions is crucial for simplifying and performing operations effectively.

Multiplying Radicals Step-by-Step

Multiplying radicals involves handling both the coefficients and the radical parts separately. Here’s how you can do it smoothly:

• First, multiply the coefficients (the numbers in front of the radicals). For example, in \(4 \sqrt{6} \times 2 \sqrt{6},\) you start by multiplying 4 and 2, giving you 8.
• Then, handle the radicals separately. Multiply the radicals using the property \(\sqrt{a} \times \sqrt{a} = \sqrt{a^2} = a.\) So, \(\sqrt{6} \times \sqrt{6} = 6.\)
• Combine your results: multiply the result from the coefficients with the result from the radicals. So, \(8 \times 6 = 48.\)
• In cases where the radicals are different, like \(\sqrt{2}\) and \(\sqrt{3},\) simply multiply them together to get \(\sqrt{6}.\)

This process helps maintain accuracy when multiplying radical expressions.

Simplifying Radical Expressions

Simplifying radical expressions involves reducing them to their simplest form while respecting their properties:

• When you encounter different radicals, as seen in \(3\sqrt{2} - 2\sqrt{3},\) remember that if the radicals aren't like terms, you cannot combine them. Keep the expression as it is.
• For multiplication, simplify step by step: Start with coefficients, then work through the radicals. If radicals multiply to form a neat square (like \(\sqrt{36}\)), write it in its simplest integer form.
• After multiplication, if you find terms that can be combined, do so to simplify the expression further.
• Finally, always double-check your work to ensure you haven't overlooked potential simplifications or combination options.

By focusing on these methods, you can keep radical expressions as simple and accurate as possible, allowing for easier operations and understanding.