Problem 24 - Algebra for College Students

Question

Find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers. \(\sqrt{2 x}(3 \sqrt{y}-7 \sqrt{5})\)

Step-by-Step Solution

Answer

The expression simplifies to \(3\sqrt{2xy} - 7\sqrt{10x}\).

Step 1: Distribute the Square Root

We start by applying the distributive property to the expression \(\sqrt{2x}(3\sqrt{y} - 7\sqrt{5})\). Distribute \(\sqrt{2x}\) to each term inside the parentheses. This gives us two separate products: \(\sqrt{2x} \cdot 3\sqrt{y}\) and \(\sqrt{2x} \cdot (-7\sqrt{5})\).

Step 2: Simplify Each Product

For the first product, \(\sqrt{2x} \cdot 3\sqrt{y}\), we multiply the coefficients and radicands separately. This becomes: \(3 \cdot \sqrt{2x} \cdot \sqrt{y} = 3\sqrt{2xy}\). For the second product, \(\sqrt{2x} \cdot (-7\sqrt{5})\), similarly multiply the coefficients and radicands: \(-7 \cdot \sqrt{2x} \cdot \sqrt{5} = -7\sqrt{10x}\).

Step 3: Write in Simplified Form

Combine both results from Step 2 into a single expression: \(3\sqrt{2xy} - 7\sqrt{10x}\). Since the terms have different radicands, they cannot be combined further. The expression is now in its simplest radical form.

Key Concepts

Distributive Property in Algebra

When you see an expression with parentheses, the distributive property is your go-to tool. This property helps you simplify expressions by multiplying a single term by each term inside the parentheses. In algebra, this is especially useful when dealing with radicals, which are numbers or expressions under a square root sign.

For example, if you're faced with an expression like \( \sqrt{2x}(3\sqrt{y}-7\sqrt{5}) \), you use the distributive property to "distribute" \( \sqrt{2x} \) across the terms within the parentheses. This turns it into two separate multiplication problems: \( \sqrt{2x} \cdot 3\sqrt{y} \) and \( \sqrt{2x} \cdot (-7\sqrt{5}) \).

• This process allows simplification by breaking down complex expressions.
• You ensure that each part of the expression is properly accounted for.

By mastering the distributive property, you lay the groundwork for other algebraic operations, making complex problems more manageable.

Multiplying Radical Expressions

Multiplying radicals might seem daunting at first, but following a clear method can simplify things substantially. When multiplying radical expressions, like \( \sqrt{2x} \cdot 3\sqrt{y} \), you handle the numbers outside the radicals (coefficients) and the numbers inside the radicals (radicands) separately.

Here's a step-by-step breakdown:

• First, multiply the coefficients. For \( 3\sqrt{y} \), the coefficient is 3.
• Next, multiply the radicands: \( \sqrt{2x} \cdot \sqrt{y} = \sqrt{2xy} \).

Continue this process for the other term: \( \sqrt{2x} \cdot (-7\sqrt{5}) \) becomes \(-7\sqrt{10x} \).

No matter the expression, this method ensures that all parts are multiplied correctly. Just be mindful of negative signs and keep radicands separate unless they can be combined.

Simplifying Radicals

Simplifying radicals is about transforming a radical expression into its simplest form without losing its value. Imagine you have the expression \( 3\sqrt{2xy} - 7\sqrt{10x} \). To simplify the radicals, you first ensure that each radical expression is written as compactly as possible.

To simplify further:

• Check if the radicand inside a square root can be simplified (like finding square factors that can be extracted).
• Combine like terms if possible – radicals with the same radicand can be added or subtracted, much like like terms with variables in algebraic expressions.

However, in this instance, \( 3\sqrt{2xy} \) and \( -7\sqrt{10x} \) have different radicands, meaning they can't be combined further. As such, the expression is already in its simplest radical form.

Simplifying radicals often requires patience and practice, but mastering these skills is rewarding and useful in various branches of mathematics.