Problem 18 - Algebra 2 and Trigonometry

Question

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers. \(\frac{4}{3-\sqrt{3}}\)

Step-by-Step Solution

Answer

The simplified form is \(2 + \frac{2\sqrt{3}}{3}\).

Step 1: Identify the Conjugate

In order to rationalize the denominator, identify the conjugate of the denominator. The denominator is \(3-\sqrt{3}\), so its conjugate is \(3+\sqrt{3}\).

Step 2: Multiply Numerator and Denominator by the Conjugate

Multiply both the numerator and the denominator by the conjugate found in Step 1. This will give us the new fraction: \(\frac{4 \cdot (3+\sqrt{3})}{(3-\sqrt{3})(3+\sqrt{3})}\).

Step 3: Simplify the Denominator

The product of conjugates \((a-b)(a+b)\) is equal to \(a^2 - b^2\). So, simplify \((3-\sqrt{3})(3+\sqrt{3})\) to \(3^2 - (\sqrt{3})^2 = 9 - 3 = 6\).

Step 4: Distribute in the Numerator

Distribute the \(4\) in the numerator: \(4 \cdot 3 + 4 \cdot \sqrt{3} = 12 + 4\sqrt{3}\).

Step 5: Write the Simplified Fraction

The rationalized fraction is \(\frac{12 + 4\sqrt{3}}{6}\). Simplify this by dividing each term in the numerator by \(6\): \(\frac{12}{6} + \frac{4\sqrt{3}}{6} = 2 + \frac{2\sqrt{3}}{3}\).

Key Concepts

Conjugates in Algebra

In algebra, the term "conjugate" refers often to a unique pairing used, especially when dealing with expressions containing square roots or complex numbers. Simply put, the conjugate of a binomial is just the original binomial with a flipped sign in between its terms. For example, if you have the expression \(a-b\), its conjugate would be \(a+b\).
Conjugates serve an important role in simplifying expressions, particularly when rationalizing denominators. When you multiply conjugates together, you get what's known as a "difference of squares". This effectively eliminates the radical (like a square root) in the denominator. This is crucial, as denominators composed of more straightforward numbers are typically considered simpler and are often much easier to work with in subsequent calculations.
By rationalizing, you're essentially moving the 'tricky' part out of the denominator and simplifying the equation greatly, setting the stage for further simplification of fractions.

Simplification of Fractions

Simplifying fractions involves reducing a fraction in such a way that both its numerator and denominator are divided by their greatest common factor. After you've rationalized the denominator, the next logical step is to simplify the overall fraction wherever possible.
In our example, after rationalizing, we obtained \(\frac{12 + 4\sqrt{3}}{6}\). The aim is to break this down further to its simplest form. Do this by dividing each term of the numerator by the denominator separately:

• Divide the whole number: \(\frac{12}{6} = 2\).
• Divide the root expression: \(\frac{4\sqrt{3}}{6} = \frac{2\sqrt{3}}{3}\).

This leaves you with a fraction that is neatly simplified, \(2 + \frac{2\sqrt{3}}{3}\), where no common factors remain apart from 1, which makes it its simplest form.

Square Roots in Algebra

Square roots, one of the fundamental concepts in algebra, often pop up in various algebraic problems, including rationalizing denominators and simplifying expressions. A square root function essentially undoes the squaring of a number. For example, \(\sqrt{9}\) equals \(3\) because \(3 \times 3 = 9\).
When square roots appear, particularly in denominators, they can make expressions much more difficult to manage. This is why algebra often focuses on rationalizing them. Rationalizing with square roots involves pairing them with their conjugates to erase the root from the denominator.
By moving the square root to the numerator (or eliminating it altogether), the expression becomes more straightforward, easing subsequent calculations and improving the readability of the algebraic expression. It's a key skill taught to make handling algebraic fractions more manageable and to prepare students for more complex mathematical topics.