Problem 11 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Combine Like Terms for Addition

Identify like terms in the expression \(4 \sqrt{6} + 2 \sqrt{6}\). Like terms have the same radical part. Here, both terms have \(\sqrt{6}\). Combine them by adding their coefficients: \(4 + 2 = 6\). Thus, \(4 \sqrt{6} + 2 \sqrt{6} = 6 \sqrt{6}\).

Step 2: Multiply the Radicals for Multiplication

Multiply the expression \(4 \sqrt{6}(2 \sqrt{6})\). Start by multiplying the coefficients: \(4 \times 2 = 8\). Next, multiply the radical terms: \(\sqrt{6} \times \sqrt{6} = \sqrt{36} = 6\). Finally, multiply the results: \(8 \times 6 = 48\). Therefore, \(4 \sqrt{6}(2 \sqrt{6}) = 48\).

Step 3: Subtraction with Different Radicals

Examine \(3 \sqrt{2} - 2 \sqrt{3}\). These are not like terms because their radical parts are different (\(\sqrt{2}\) and \(\sqrt{3}\)). So, this expression cannot be simplified further. It remains as \(3 \sqrt{2} - 2 \sqrt{3}\).

Step 4: Multiply Radicals with Different Bases

For the operation \(3 \sqrt{2}(-2 \sqrt{3})\), begin by multiplying the coefficients: \(3 \times -2 = -6\). Next, multiply the radicals: \(\sqrt{2} \times \sqrt{3} = \sqrt{6}\). Combine to get: \(3 \sqrt{2}(-2 \sqrt{3}) = -6 \sqrt{6}\).

Key Concepts

Combining Like Terms

In math, like terms are terms that have identical variables and powers. For radical expressions, like terms have the same radicand, which is the number under the radical sign. So when combining like terms in radicals such as \(4 \sqrt{6} + 2 \sqrt{6}\), it's just like adding similar algebraic terms. Since both terms have \(\sqrt{6}\) as their radical part, you only need to add the coefficients — which are the numbers in front of the radicals.
Here's how you do it:

• Identify terms with the same radical part.
• Add the coefficients (the numbers before the radical).

For the given example, this would be \(4 + 2 = 6\). Then, put \(\sqrt{6}\) after that sum: \(6 \sqrt{6}\). As you can see, combining like terms can greatly simplify the expression.

Multiplying Radicals

When multiplying radicals, such as in \(4 \sqrt{6}(2 \sqrt{6})\), the process involves multiplying both the coefficients and the radicands. This helps in breaking down the complex process into simpler steps.
Let's see it step-by-step:

• First, multiply the coefficients: \(4 \times 2 = 8\).
• Then, multiply the radicands: \(\sqrt{6} \times \sqrt{6} = \sqrt{36} = 6\).
• Finally, multiply the products obtained from the above two steps: \(8 \times 6 = 48\).

So \(4 \sqrt{6}(2 \sqrt{6}) = 48\). Multiplying two identical radicals results in a whole number because the square root of a number times itself is the number inside the radical. Understanding this will make these calculations seamless.

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form. In cases like \(3 \sqrt{2} - 2 \sqrt{3}\), where the radicals are different, simplification of the expression by combining terms isn't possible. This is because the radicands, \(\sqrt{2}\) and \(\sqrt{3}\), are not the same, and thus, are unlike terms.
Here's what to note:

• When radicands differ, you cannot perform direct addition or subtraction. Hence, it stays as \(3 \sqrt{2} - 2 \sqrt{3}\).
• However, if multiplying radicals with different bases, like in \(3 \sqrt{2}(-2 \sqrt{3})\), you handle the coefficients and radicands separately.
• Multiply the coefficients: \(3 \times -2 = -6\).
• Multiply the radicands: \(\sqrt{2} \times \sqrt{3} = \sqrt{6}\).
• Join them into \(-6 \sqrt{6}\).

Even when the radicals don't combine, multiplication links them in a new, simpler way.