Problem 37 - Algebra for College Students

Question

Express each of the following in simplest radical form. All variables represent positive real numbers. \(\sqrt{\frac{2 x}{5 y}}\)

Step-by-Step Solution

Answer

Simplified form: \( \frac{\sqrt{2x}}{\sqrt{5y}} \); or optionally \( \frac{\sqrt{10xy}}{5y} \).

Step 1: Review the Given Expression

We need to simplify the expression \( \sqrt{\frac{2x}{5y}} \). This expression involves taking the square root of a fraction.

Step 2: Apply the Square Root Rule

Recall that the square root of a fraction \( \frac{a}{b} \) can be expressed as \( \frac{\sqrt{a}}{\sqrt{b}} \). Apply this rule to the expression: \[ \sqrt{\frac{2x}{5y}} = \frac{\sqrt{2x}}{\sqrt{5y}} \].

Step 3: Simplify Each Square Root Separately

Consider the expression \( \frac{\sqrt{2x}}{\sqrt{5y}} \). We need to handle \( \sqrt{2x} \) and \( \sqrt{5y} \) separately. For \( \sqrt{2x} \), it can remain as it is since 2 and x are both under the square root. Similarly, for \( \sqrt{5y} \), 5 and y remain under their square root.

Step 4: Rationalize the Denominator (if necessary)

The expression \( \frac{\sqrt{2x}}{\sqrt{5y}} \) is already simplified. However, if you want to represent it without a square root in the denominator, multiply the numerator and denominator by \( \sqrt{5y} \): \[ \frac{\sqrt{2x} \cdot \sqrt{5y}}{\sqrt{5y} \cdot \sqrt{5y}} = \frac{\sqrt{10xy}}{5y} \]. This is an optional step depending on the desired form of the expression.

Key Concepts

Square Root Rule

When it comes to simplifying radicals, especially those involving fractions, the square root rule is an essential tool. This rule states that the square root of a fraction, \(\sqrt{\frac{a}{b}}\), can be separated into the square root of the numerator and the square root of the denominator, resulting in \(\frac{\sqrt{a}}{\sqrt{b}}\). This separation allows us to simplify each part individually. For instance, in the expression \(\sqrt{\frac{2x}{5y}}\), applying the square root rule gives us \(\frac{\sqrt{2x}}{\sqrt{5y}}\). This makes things more manageable as it reduces a complex radical into smaller, simpler parts by breaking them down into more familiar components.
When performing this step, make sure that \(b\), the value in the denominator, is not zero to avoid division by zero issues. Moreover, when dealing with variables within these components, always assume they are positive to ensure validity of the operations performed.

Fraction Division

Fraction division in radical expressions involves dividing two numbers or expressions written in fraction form. After applying the square root rule, you often end up with an expression like \(\frac{\sqrt{2x}}{\sqrt{5y}}\). At this stage, each square root is treated as a separate entity.
To simplify the division, we observe each component for further simplification, such as simplifying individual radicals if possible. For the numerator \(\sqrt{2x}\), you are required to find square factors of 2 and x, if any, which isn't applicable here as neither 2 nor x has a perfect square factor other than 1. Similarly, you'll perform the same inspection for the denominator \(\sqrt{5y}\).
Each expression is examined separately and individually simplified before they are brought back together. If direct simplification isn't possible, this state is considered as the simplest form. But always keep in mind that maintaining the integrity of positive values ensures accuracy.

Rationalizing the Denominator

In mathematics, the process of rationalizing the denominator involves eliminating any radicals present in the denominator of a fraction. This ensures the denominator is a rational number. For instance, starting with \(\frac{\sqrt{2x}}{\sqrt{5y}}\), if you desire a form without radicals at the base, you multiply both the numerator and the denominator by \(\sqrt{5y}\).
This leads to:

• Nominator: \(\sqrt{2x} \times \sqrt{5y} = \sqrt{10xy}\)
• Denominator: \(\sqrt{5y} \times \sqrt{5y} = 5y\)

Thus, the expression becomes \(\frac{\sqrt{10xy}}{5y}\). This step not only helps in simplifying but also in standardizing expressions to a form that is usually preferred in mathematical contexts. Rationalizing ensures consistent handling when performing algebraic operations, and it's easier to compare such expressions across different equations.