Problem 24 - Intermediate Algebra

Question

Rationalize each denominator. See Examples 1 through 3. $$ \sqrt{\frac{11 y}{45}} $$

Step-by-Step Solution

Answer

\(\frac{\sqrt{495y}}{45}\)

Step 1: Simplify the Expression

Start by rewriting the expression \[\sqrt{\frac{11y}{45}} = \frac{\sqrt{11y}}{\sqrt{45}}.\]This setup allows us to separately handle the radical in the denominator.

Step 2: Rationalize the Denominator

To rationalize the denominator, we need to remove the square root. The simplest approach is to multiply both the numerator and denominator by \(\sqrt{45}\):\[\frac{\sqrt{11y}}{\sqrt{45}} \times \frac{\sqrt{45}}{\sqrt{45}} = \frac{\sqrt{11y} \cdot \sqrt{45}}{45}.\]This eliminates the square root from the denominator.

Step 3: Simplify the Numerator

In the numerator, use the property that \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\):\[\sqrt{11y} \cdot \sqrt{45} = \sqrt{11y \times 45} = \sqrt{495y}.\]

Step 4: Write the Final Expression

The expression is now rationalized:\[\frac{\sqrt{495y}}{45}.\]This is the simplified and rationalized form of the original expression.

Key Concepts

Understanding Radical Expressions

Radical expressions are mathematical expressions that contain a square root, cube root, or any other form of root. They often involve terms under a radical sign, also known as the square root symbol (√). The expression shown in the original exercise is a radical expression, specifically a square root:

• Radicand: The number or expression inside the radical sign, for example, in \(\sqrt{11y}\), 11y is the radicand.
• Index: While oftentimes not shown, the index of a square root is 2, signifying the square root. For a cube root, it would be 3 and so on.

Radical expressions can sometimes be complex due to the presence of roots. Understanding each component, like the radicand and index, simplifies the process of handling these expressions. This is the foundation that supports operations such as rationalizing denominators.

Simplifying Radicals

Simplifying radicals involves breaking down the radicand (the number under the square root) into its factors to make it more straightforward. This is an important skill when dealing with rational expressions, as it eases other calculations.

• Factor the radicand into prime multiples. For example, in \(\sqrt{45}\), factor 45 into its prime factors which yields \(3 \times 3 \times 5\).
• Group the factors to form pairs, which can be pulled out from under the radical. Thus, \(\sqrt{45} = \sqrt{3^2 \times 5} = 3\sqrt{5}\).

Simplifying radicals often aids in reducing the complexity of the expression, making subsequent steps such as rationalization more manageable.

Properties of Square Roots

Square roots come with specific properties that are useful when working with radical expressions, especially when simplifying or rationalizing denominators.

• Product Property: The square root of a product equals the product of the square roots, \(\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}\). This property simplifies operations involving products under a square root.
• Quotient Property: The square root of a quotient equals the quotient of the square roots, \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\), assuming \(b eq 0\). This property helps rationalize denominators by separating the radicals in the numerator and denominator.
• Non-negative Property: The square root of a number is non-negative, which means \(\sqrt{x}\geq 0\) for \(x\geq 0\).

Understanding these properties helps in simplifying radical expressions and plays a crucial role in operations like rationalizing the denominator of fractions with radicals. By applying these properties correctly, complex expressions can be transformed into simpler, more manageable forms.