Problem 42 - Algebra A Combined Function

Question

Rationalize each denominator. Assume that all variables represent positive real numbers. \(\frac{2 \sqrt{a}}{2 \sqrt{x}-\sqrt{y}}\)

Step-by-Step Solution

Answer

The rationalized expression is \( \frac{4\sqrt{ax} + 2\sqrt{ay}}{4x - y} \).

Step 1: Identifying the Problem

We need to rationalize the denominator of the expression \( \frac{2 \sqrt{a}}{2 \sqrt{x}-\sqrt{y}} \). Rationalizing the denominator involves eliminating any radicals in the denominator by multiplying the numerator and denominator by an expression that will achieve this.

Step 2: Multiplying by Conjugate

To rationalize \( 2\sqrt{x} - \sqrt{y} \), we multiply the fraction by the conjugate of the denominator, which is \( 2\sqrt{x} + \sqrt{y} \), both in the numerator and denominator. The expression becomes: \( \frac{2 \sqrt{a} (2\sqrt{x} + \sqrt{y})}{(2 \sqrt{x} - \sqrt{y})(2 \sqrt{x} + \sqrt{y})} \).

Step 3: Expanding the Denominator

Use the identity \((a-b)(a+b) = a^2 - b^2\) to eliminate the radicals in the denominator: \( \big(2 \sqrt{x}\big)^2 - \big(\sqrt{y}\big)^2 = 4x - y \).

Step 4: Expanding the Numerator

Distribute \(2\sqrt{a}\) in the numerator: \(2\sqrt{a} \cdot 2\sqrt{x} + 2\sqrt{a} \cdot \sqrt{y} = 4\sqrt{ax} + 2\sqrt{ay}\).

Step 5: Writing the Rationalized Expression

The fully rationalized expression is \( \frac{4\sqrt{ax} + 2\sqrt{ay}}{4x - y} \). This expression now has a rational denominator.

Key Concepts

Conjugate in Algebra

In algebra, conjugates are pairs of expressions that differ only by the sign between two terms. Conjugates are useful because their product almost always results in a rational number, which means it will not contain any radicals. Consider a binomial expression like \[ a + b \] and its conjugate \[ a - b \].

• When you multiply these expressions, you get \[ (a+b)(a-b) = a^2 - b^2 \], which is the difference of squares formula.
• This formula helps in eliminating radicals or complex numbers when rationalizing fractions.

To rationalize a denominator with radicals, we often multiply both the numerator and the denominator by the conjugate of the denominator. This process eliminates the radical from the denominator, making it rational.

Radicals in Algebra

Radicals in algebra represent the root of a number. The most common radical is the square root, denoted by a symbol that looks like a checkmark with a horizontal line over the numbers or variables whose root we want to find. Here's an essential guide:

• A square root \( \sqrt{x} \) refers to a number which when multiplied by itself gives back \( x \).
• This concept extends to cube roots, fourth roots, etc., where the radical symbol, \( \sqrt[3]{y} \) or \( \sqrt{straight}{x}\), varies.
• In expressions, radicals often appear with coefficients like \( 2\sqrt{x} \) or variables within them, such as \( \sqrt{y+a} \).

Rationalizing techniques often require the manipulation of radicals using conjugates, allowing expressions to be simplified in ways that ensure denominators are free of radicals, thus making further computation manageable.

Distributive Property in Algebra

The distributive property in algebra helps you expand expressions. It states that multiplying a number by a sum (or difference) is the same as multiplying the number by each term within the parentheses and then adding (or subtracting) the products. For example: \[ a(b+c) = ab + ac \].

• This property is particularly useful when dealing with rational expressions that need simplification.
• In our exercise example, distributing \( 2\sqrt{a} \) across \( (2\sqrt{x} + \sqrt{y}) \) involves multiplying \( 2\sqrt{a} \) with each term inside the parentheses.
• This results in \( 4\sqrt{ax} + 2\sqrt{ay} \), showing how the distributive property facilitates the expansion of expressions to aid in simplification.

By expanding expressions in this way, we work towards obtaining a simplified form where all terms are adequately expressed, making the process of rationalizing denominators straightforward.