Problem 81 - Algebra and Trigonometry

Question

\(77-82\) me Rationalize the denominator. $$ \frac{y}{\sqrt{3}+\sqrt{y}} $$

Step-by-Step Solution

Answer

Rationalized expression: \( \frac{y\sqrt{3} - y\sqrt{y}}{3 - y} \).

Step 1: Identify the Conjugate

To rationalize the denominator \( \sqrt{3} + \sqrt{y} \), we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \( \sqrt{3} + \sqrt{y} \) is \( \sqrt{3} - \sqrt{y} \).

Step 2: Multiply by the Conjugate

Multiply both the numerator and the denominator by \( \sqrt{3} - \sqrt{y} \):\[\frac{y}{\sqrt{3} + \sqrt{y}} \times \frac{\sqrt{3} - \sqrt{y}}{\sqrt{3} - \sqrt{y}} = \frac{y(\sqrt{3} - \sqrt{y})}{(\sqrt{3} + \sqrt{y})(\sqrt{3} - \sqrt{y})}.\]

Step 3: Simplify the Denominator

Apply the difference of squares formula to the denominator:\[(\sqrt{3} + \sqrt{y})(\sqrt{3} - \sqrt{y}) = (\sqrt{3})^2 - (\sqrt{y})^2 = 3 - y.\]So, the expression becomes:\[\frac{y(\sqrt{3} - \sqrt{y})}{3 - y}.\]

Step 4: Simplify the Numerator

Distribute \( y \) in the numerator:\[\frac{y(\sqrt{3}) - y(\sqrt{y})}{3 - y} = \frac{y\sqrt{3} - y\sqrt{y}}{3 - y}.\]The expression is now fully rationalized.

Key Concepts

Conjugate in Algebra

The concept of a conjugate in algebra is a fundamental tool used in simplifying expressions, especially when dealing with radicals. A conjugate refers to a binomial expression formed by changing the sign of the second term. For the expression \( \sqrt{3} + \sqrt{y} \), its conjugate is \( \sqrt{3} - \sqrt{y} \). Using conjugates helps in eliminating the square root from the denominator.

When you're given a radical expression, the goal of using the conjugate is to create a difference of squares situation. This happens because multiplying conjugates always results in a rational expression. As seen in the exercise, multiplying \( \sqrt{3} + \sqrt{y} \) with \( \sqrt{3} - \sqrt{y} \) involves the difference of these terms' squares.

Difference of Squares

The difference of squares is a handy algebraic identity given by \((a + b)(a - b) = a^2 - b^2\). This formula is vital when working with conjugates since it allows us to simplify the denominator easily.

In the problem, multiplying \( (\sqrt{3} + \sqrt{y})(\sqrt{3} - \sqrt{y}) \) fits this pattern. Applying the difference of squares results in:

• \((\sqrt{3})^2 - (\sqrt{y})^2\)
• Which simplifies to \(3 - y\)

This technique transforms complex expressions with radicals into simpler, more manageable forms by eliminating the square root in the denominator.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves manipulating and rewriting expressions into a more understandable or usable form. This includes using operations like distributing multiplication over addition or subtraction, and simplifying fractions.

In the exercise, once the denominator is rationalized, the next step involves simplifying the numerator. The numerator \(y(\sqrt{3} - \sqrt{y})\) is expanded by distributing \(y\) to each term:

• Resulting in \(y\sqrt{3} - y\sqrt{y}\)

Now both the numerator and denominator form a rational expression \(\frac{y\sqrt{3} - y\sqrt{y}}{3 - y}\), making the expression cleaner and easier to work with. Each step brings us closer to simplifying complex rational expressions into simpler forms that are advantageous in algebraic manipulations.