Question

Rationalize the denominator of each expression. Assume that all variables are positive when they appear. $$\frac{2-\sqrt{5}}{2+3 \sqrt{5}}$$

Short answer

-\frac{19}{41} + \frac{8 \sqrt{5}}{41}

Step-by-step solution

Identify the conjugate

The denominator is \(2 + 3 \sqrt{5}\). The conjugate of this expression is \(2 - 3 \sqrt{5}\). To rationalize the denominator, multiply both the numerator and the denominator by this conjugate.

Multiply numerator and denominator by the conjugate

Multiply both the numerator and the denominator of the fraction by \(2 - 3 \sqrt{5}\): \[\frac{(2 - \sqrt{5})(2 - 3 \sqrt{5})}{(2 + 3 \sqrt{5})(2 - 3 \sqrt{5})}\]

Expand the numerator

Use the distributive property (FOIL method) to expand the numerator: \[ (2 - \sqrt{5})(2 - 3 \sqrt{5}) = 2 \cdot 2 + 2 \cdot (-3 \sqrt{5}) - \sqrt{5} \cdot 2 - \sqrt{5} \cdot (-3 \sqrt{5}) \] Simplifying this: \[ 4 - 6 \sqrt{5} - 2 \sqrt{5} + 3 \cdot 5 \] Combine like terms: \[ 4 - 8 \sqrt{5} + 15 = 19 - 8 \sqrt{5} \]

Simplify the denominator

Use the difference of squares formula to simplify the denominator: \[ (2 + 3 \sqrt{5})(2 - 3 \sqrt{5}) = 2^2 - (3 \sqrt{5})^2 \] This simplifies to: \[ 4 - 45 = -41 \]

Write the final answer

Combine the simplified numerator and denominator: \[ \frac{19 - 8 \sqrt{5}}{-41} \] This can further be written as: \[ -\frac{19}{41} + \frac{8 \sqrt{5}}{41} \]

Key concepts

Conjugates

When working to rationalize the denominator, understanding conjugates is essential. A conjugate pairs with a binomial expression to eliminate square roots in the denominator.
Conjugates look like this: if you have an expression of the form \(a + b\), its conjugate is \(a - b\), and vice versa.
In our example, the denominator is \(2 + 3\sqrt{5}\). Therefore, the conjugate is \(2 - 3\sqrt{5}\).
By multiplying the numerator and the denominator by this conjugate, we can eliminate the irrational part from the denominator.

Difference of Squares

The difference of squares is a useful algebraic identity that simplifies expressions involving conjugates.
The identity is \(a^2 - b^2 = (a+b)(a-b)\). This formula helps in reducing the product of the original number and its conjugate to simpler terms.
For our denominator \((2 + 3 \sqrt{5})(2 - 3 \sqrt{5})\):
We can write it as \(2^2 - (3 \sqrt{5})^2\).
Calculate these squares to get \(4 - 45 = -41\).
This step is crucial for rationalizing the denominator, rendering the fraction more manageable.

FOIL Method

To multiply the numerator by the conjugate, the FOIL method comes in handy.
FOIL stands for First, Outer, Inner, Last and is used to expand binomials like \((2 - \sqrt{5})(2 - 3 \sqrt{5})\).
Here’s the step-by-step breakdown:

• First: \(2 \cdot 2 = 4\)
• Outer: \(2 \cdot (-3 \sqrt{5}) = -6 \sqrt{5}\)
• Inner: \(-\sqrt{5} \cdot 2 = -2 \sqrt{5}\)
• Last: \(-\sqrt{5} \cdot (-3 \sqrt{5}) = 15\)

Combining these, we get: \(4 - 6 \sqrt{5} - 2 \sqrt{5} + 15 = 19 - 8 \sqrt{5}\).
This final expanded form helps in further steps of simplification.

Simplifying Fractions

After expanding and combining like terms, we often get a compound fraction like \( \frac{19 - 8 \sqrt{5}}{-41} \).
To simplify, break it into smaller parts and reformat it:

• First, separate each term in the numerator by the denominator: \(-\frac{19}{41} + \frac{8 \sqrt{5}}{41}\)

By breaking down the fraction this way, we make it much simpler to work with and understand. Reformatting into individual simpler fractions enhances clarity, especially when dealing with complex terms involving square roots.