Problem 23 - Algebra for College Students

Question

Find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers. \(3 \sqrt{x}(5 \sqrt{2}+\sqrt{y})\)

Step-by-Step Solution

Answer

The product is \(15 \sqrt{2x} + 3 \sqrt{xy}\).

Step 1: Distribute the First Term

Start by distributing the term \(3\sqrt{x}\) to each term inside the parentheses. This means you will multiply \(3\sqrt{x}\) by both \(5\sqrt{2}\) and \(\sqrt{y}\).

Step 2: Multiply with the First Term inside the Parentheses

Multiply \(3\sqrt{x}\) by \(5\sqrt{2}\). This gives: \[3 \sqrt{x} \cdot 5 \sqrt{2} = 15 \sqrt{2x}.\]

Step 3: Multiply with the Second Term inside the Parentheses

Next, multiply \(3\sqrt{x}\) by \(\sqrt{y}\). This gives: \[3 \sqrt{x} \cdot \sqrt{y} = 3 \sqrt{xy}.\]

Step 4: Combine the Results

Combine the results from the multiplication steps to get the full expression: \[15 \sqrt{2x} + 3 \sqrt{xy}.\] This is the expression in simplest radical form.

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental mathematical principle that allows you to multiply a single term by each term within parentheses. This is crucial when working with expressions that involve adding or subtracting terms. In this exercise, we use the distributive property to expand the expression \(3\sqrt{x}(5\sqrt{2}+\sqrt{y})\).

Here's how it works:

• We take the term outside the parentheses, \(3\sqrt{x}\), and multiply it by each term inside the parentheses one at a time.
• First, multiply \(3\sqrt{x}\) by \(5\sqrt{2}\).
• Then, multiply \(3\sqrt{x}\) by \(\sqrt{y}\).

Each multiplication gives you a part of the expanded expression, which you then add together to form the final expression. This step-by-step approach ensures that you properly distribute the multiplier over all terms inside the parentheses.

The Art of Multiplying Radicals

When multiplying radicals, it's important to remember that the coefficients (numbers outside the square root) are multiplied separately from the radicands (numbers inside the square root). In our example, \(3\sqrt{x} \cdot 5\sqrt{2}\) and \(3\sqrt{x} \cdot \sqrt{y}\), we can break this down further.

For the first multiplication:

• Multiply the coefficients \(3\) and \(5\) to get \(15\).
• Multiply the radicands \(x\) and \(2\) to get \(2x\).
• Combine them under one square root: \(\sqrt{2x}\).

Resulting in the product \(15\sqrt{2x}\).

For the second multiplication:

• The coefficient \(3\) is carried over as it is.
• Multiply radicands \(x\) and \(y\) to get \(xy\).
• This gives us the expression \(3\sqrt{xy}\).

These steps allow you to maintain the structure of the radical expressions while simplifying them efficiently. Understanding how to handle each component separately is key to mastering this operation.

Simplifying Square Roots

Once we have multiplied and combined our expressions into \(15\sqrt{2x} + 3\sqrt{xy}\), we need to check if they can be simplified further. Here are some general tips for simplifying square roots, which you can apply:

• Look for perfect square factors within the radicand. If you find any, factor them out.
• Separate the perfect square from the non-square part inside the square root and simplify it.
• Simplify radicals by checking if any common factors exist.

In our result, neither \(\sqrt{2x}\) nor \(\sqrt{xy}\) has perfect square factors that can be further simplified. This means the expressions are in their simplest form.

Simplifying radicals often involves some trial and error, but with practice, it becomes easier to spot simplifications quickly. Ensuring the expression is fully simplified allows you to work with a cleaner form of the radical, which is essential for further mathematical operations.