Problem 25 - Algebra for College Students

Question

Find the following products and express answers in simplest radical form. All variables represent nonnegative real numbers. \(\sqrt{x y}(5 \sqrt{x y}-6 \sqrt{x})\)

Step-by-Step Solution

Answer

The expression simplifies to \(5xy - 6x \sqrt{y}\).

Step 1: Expand the Expression

Start by distributing \(\sqrt{xy}\) to each term inside the parentheses in the expression: \(\sqrt{xy}(5 \sqrt{xy} - 6 \sqrt{x})\). This gives us two separate expressions: \((\sqrt{xy})(5 \sqrt{xy}) - (\sqrt{xy})(6 \sqrt{x})\).

Step 2: Simplify Each Term

First, consider \(\sqrt{xy} \cdot 5\sqrt{xy}\). Use the property \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\) to simplify this to \[5 \cdot \sqrt{xy \cdot xy} = 5 \sqrt{(xy)^2} = 5xy\].Next, simplify \(\sqrt{xy} \cdot 6\sqrt{x}\) to \[6 \cdot \sqrt{xy \cdot x} = 6 \sqrt{x^2y} = 6x \sqrt{y}\].

Step 3: Write the Completed Expression

Upon simplifying both parts, rewrite the full product expression: \((\sqrt{xy})(5 \sqrt{xy}) - (\sqrt{xy})(6 \sqrt{x}) = 5xy - 6x \sqrt{y}\). This gives us the final answer.

Key Concepts

Product of Radicals

When dealing with the product of radicals, you must remember the key property: \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \). This property allows you to multiply the radicands (the numbers inside the square root sign) together directly under a single square root.
However, there are a few simple rules to keep in mind:

• You can only apply this rule when both radicands are under square roots.
• Always ensure that \( a \) and \( b \) are non-negative to avoid undefined results.

For example, if you have \( \sqrt{xy} \cdot \sqrt{xy} \), you'd multiply the radicands: \( xy \times xy = (xy)^2 \), resulting in \( \sqrt{(xy)^2} \), which simplifies to \( xy \).
Understanding this basic property is crucial when working with more complex radical expressions, like multiplying a single term by a binomial radical.

Distributive Property

The distributive property is a critical concept in algebra that allows you to multiply a term across terms inside parentheses. This property is succinctly expressed as \( a(b + c) = ab + ac \).
To distribute, take each term inside the parentheses and multiply it by the term outside. This is exactly what we do when we expand expressions like \( \sqrt{xy}(5\sqrt{xy} - 6\sqrt{x}) \).

• The term \( \sqrt{xy} \) is distributed first to \( 5\sqrt{xy} \), giving \( 5\sqrt{xy \cdot xy} \).
• Next, it is distributed to \( -6\sqrt{x} \), resulting in \( -6\sqrt{xy \cdot x} \).

By distributing the terms correctly, we transform the original expression into separate simpler terms, paving the way for further simplification.
This simplification is crucial as it breaks down complex expressions into parts that are easier to handle and solve.

Simplifying Radical Expressions

Simplifying radical expressions involves reducing them to their most basic form, known as the simplest radical form. This ensures that all expressions appear as simplified as possible, making them easier to understand and work with.
The process generally includes a few key steps:

• Combine like terms under the same radical when possible.
• Convert the product of identical radicals into its simplest form, such as turning \( \sqrt{(xy)^2} \) into \( xy \)
• Simplify further by identifying and removing perfect squares in the radicand.

For instance, after distributing \( \sqrt{xy} \) in the example expression, you get terms like \( 5xy \) and \( 6x \sqrt{y} \). Both are simplified forms as they can no longer be reduced further under radicals and represent the simplest form of the product.
Mastering the art of simplifying radical expressions not only makes algebra easier but also reinforces understanding of how radicals work in mathematical expressions.