Problem 21 - Algebra 2 and Trigonometry

Question

Rationalize the denominator and write each fraction in simplest form. All variables represent positive numbers. \(\frac{3}{\sqrt{5}-2}\)

Step-by-Step Solution

Answer

\(3\sqrt{5} + 6\) is in simplest form after rationalizing the denominator.

Step 1: Identify Conjugate

To rationalize the denominator of \( \frac{3}{\sqrt{5}-2} \), identify the conjugate of the denominator, which is \( \sqrt{5}+2 \). The conjugate is found by changing the sign between the two terms in the denominator.

Step 2: Multiply by the Conjugate

Multiply both the numerator and the denominator by the conjugate, \( \sqrt{5}+2 \). This gives: \( \frac{3(\sqrt{5}+2)}{(\sqrt{5}-2)(\sqrt{5}+2)} \).

Step 3: Apply the Difference of Squares

Use the difference of squares formula on the denominator: \((\sqrt{5})^2 - (2)^2\). Simplify it to \(5 - 4 = 1\). So the denominator becomes 1.

Step 4: Simplify the Numerator

Distribute the 3 in the numerator: \(3 \times \sqrt{5} + 3 \times 2 = 3\sqrt{5} + 6\).

Step 5: Divide the Terms

Since the denominator simplifies to 1, the fraction becomes \(3\sqrt{5} + 6\). The expression is already in its simplest form as there are no common factors to factor out.

Key Concepts

Conjugate

When faced with a denominator containing a radical expression, rationalizing it helps to simplify the fraction. The concept of a "conjugate" comes into play. A conjugate in mathematics is simply a pair of binomials with the same two terms, but opposite operations—a plus becomes a minus, or vice versa.
For example, if you have a denominator like \( \sqrt{5} - 2 \), the conjugate would be \( \sqrt{5} + 2 \).
This change helps in the multiplication process to eliminate the radicals from the denominator. Why? Because when you multiply a binomial by its conjugate, you get a difference of squares—a concept that simplifies nicely.

Difference of Squares

The difference of squares is a specific algebraic identity: \( a^2 - b^2 = (a - b)(a + b) \). It’s a crucial tool for rationalizing denominators that involve conjugates.
When you multiply \( (\sqrt{5} - 2) \) by its conjugate \( (\sqrt{5} + 2) \), this formula is applied:

• \( (\sqrt{5})^2 - (2)^2 = 5 - 4 \)
• The result is 1, a whole number.

This simplifies calculations significantly since it transforms a potentially complicated expression into something straightforward—radical-free and easy to work with.

Simplifying Fractions

Once you've used the conjugate and applied the difference of squares, you're left with simplifying the fraction. In the given problem, the denominator turns into 1, which makes things even easier.
Now, you just need to focus on the numerator:\( 3(\sqrt{5} + 2) \) becomes \( 3\sqrt{5} + 6 \) after distributing multiplication.
Since the denominator is 1, the expression stays the same after dividing, meaning our job here is done. Your original fraction \( \frac{3}{\sqrt{5} - 2} \) cleanly morphs into \( 3\sqrt{5} + 6 \) once it's been rationalized and simplified.