Problem 93 - Algebra and Trigonometry

Question

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 Rationalization Technique

To rationalize the denominator of the expression \( \frac{y}{\sqrt{3} + \sqrt{y}} \), we will 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 the conjugate \( \sqrt{3} - \sqrt{y} \): \[\frac{y}{\sqrt{3} + \sqrt{y}} \times \frac{\sqrt{3} - \sqrt{y}}{\sqrt{3} - \sqrt{y}} = \frac{y(\sqrt{3} - \sqrt{y})}{(\sqrt{3} + \sqrt{y})(\sqrt{3} - \sqrt{y})}.\]

Step 3: Simplify the Denominator

Use the difference of squares formula \((a+b)(a-b) = a^2 - b^2\) to simplify the denominator: \[(\sqrt{3})^2 - (\sqrt{y})^2 = 3 - y.\]

Step 4: Simplify the Numerator

Distribute \(y\) across the terms in the conjugate: \[y(\sqrt{3} - \sqrt{y}) = y\sqrt{3} - y\sqrt{y}.\]

Step 5: Write the Rationalized Expression

Combine the simplified numerator and denominator: \[\frac{y\sqrt{3} - y\sqrt{y}}{3 - y}.\] This is the expression with the rationalized denominator.

Key Concepts

Difference of Squares

The difference of squares is a mathematical technique used to simplify expressions that involve two squared terms with a subtraction sign between them. It follows the formula

• \[(a+b)(a-b) = a^2 - b^2\].

This formula states that the product of a binomial and its conjugate results in the difference of their squares. It allows us to eliminate the square roots in the denominator when rationalizing.
In our example, when rationalizing

• \[ \frac{y}{\sqrt{3} + \sqrt{y}} \],

we use its conjugate

• \[ \sqrt{3} - \sqrt{y} \]

for multiplication. The application of the difference of squares formula simplifies the denominator from a mixture of roots to a pure difference, calculated as

• \[ 3 - y \].

This simplification process is invaluable for removing irrational numbers from the denominator, a common requirement in algebra.

Conjugate

The conjugate of a binomial expression involving square roots is simply the expression with the sign between the terms flipped. So, if you have

• \[ a+b \],

its conjugate is

• \[ a-b \].

This concept is crucial when you want to rationalize denominators. The reason being, multiplying a binomial by its conjugate helps remove square roots, thanks to the difference of squares formula.
In practical terms, when rationalizing

• \[ \frac{y}{\sqrt{3} + \sqrt{y}} \],

you use its conjugate

• \[ \sqrt{3} - \sqrt{y} \]

to multiply both the numerator and the denominator. This action keeps the expression equivalent but changes its form, simplifying the denominator into a rational expression while also transforming the expression in a way that makes it easier to work with.

Simplifying Expressions

Simplifying expressions is an essential skill in algebra which involves rewriting expressions in a more manageable form. This often includes reducing the number of operations, cancelling terms, or combining like terms. When dealing with expressions that have irrational roots in the denominator, simplifying often involves rationalizing the denominator. This makes the expression easier to manage, particularly in further calculations or when applying limits.
In the case of

• \[ \frac{y}{\sqrt{3} + \sqrt{y}} \],

simplification begins by using the conjugate to rationalize the denominator, transforming it with the difference of squares into \(3 - y\). The numerator is simplified by distributing \(y\) across the conjugate, yielding

• \[ y\sqrt{3} - y\sqrt{y} \].

These steps combined give a final expression

• \[ \frac{y\sqrt{3} - y\sqrt{y}}{3 - y} \],

which is simpler to handle and free from any irrational numbers in the denominator.