Question

Answers are given at the end of these exercises. Rationalize the denominator of \(\frac{3}{2+\sqrt{3}}\).

Short answer

6 - 3\sqrt{3}

Step-by-step solution

- Identify the Conjugate

To rationalize the denominator of a fraction where the denominator is in the form of \(a + b\sqrt{c}\), the conjugate is: \(a - b\sqrt{c}\). In this exercise, the denominator is \(2 + \sqrt{3}\), so the conjugate is \(2 - \sqrt{3}\).

- Multiply the Numerator and Denominator by the Conjugate

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

- Simplify the Numerator

Distribute the 3 in the numerator: \[3(2 - \sqrt{3}) = 3 \cdot 2 - 3 \cdot \sqrt{3} = 6 - 3\sqrt{3} \]

- Simplify the Denominator

Use the difference of squares formula to simplify the denominator: \[(2 + \sqrt{3})(2 - \sqrt{3}) = 2^2 - (\sqrt{3})^2 = 4 - 3 = 1 \]

- Combine Results

Combine the results from Step 3 and Step 4: \[\frac{6 - 3\sqrt{3}}{1} = 6 - 3\sqrt{3} \]

Key concepts

Conjugate

To understand rationalizing the denominator, you must first understand the concept of the conjugate. The conjugate of a number in the format \(a + b\sqrt{c}\) is \(a - b\sqrt{c}\). Here, the terms \(a\) and \(b\) are numbers, and \(c\) is a positive integer inside a square root. The key idea is that by multiplying a number by its conjugate, we can eliminate the square root in the denominator. This simplifies our expression.

Difference of Squares

The difference of squares formula is a handy mathematical tool used when dealing with conjugates. It states: \[ (a + b)(a - b) = a^2 - b^2 \] In our exercise, this formula helps us simplify the denominator. For instance, if our denominator is \(2 + \sqrt{3}\), the difference of squares tells us that: \[ (2 + \sqrt{3})(2 - \sqrt{3}) = 2^2 - (\sqrt{3})^2 \] This further simplifies to: \[ 4 - 3 = 1 \] By using the conjugate and the difference of squares, we rationalize the denominator and make our fraction simpler.

Fractions

A fraction is a way to express a division between two numbers or expressions. It has two parts: the numerator (top) and the denominator (bottom). In some cases, the denominator contains a square root, making it hard to handle. Rationalizing the denominator makes fractions easier to work with. In our example, we started with the fraction \( \frac{3}{2+\sqrt{3}} \). Multiplying by the conjugate \( 2-\sqrt{3} \) helps us clear the square root from the denominator.

Simplifying Expressions

Simplifying expressions means making them into their simplest form. In our rationalization process, we simplify both the numerator and the denominator:

• First, we use the distributive property to expand the terms.
• Second, we apply the difference of squares formula to simplify the denominator.

For the fraction \( \frac{3}{2+\sqrt{3}} \), our steps were:

• Multiplying numerator and denominator by the conjugate: \( \frac{3(2-\sqrt{3})}{(2+\sqrt{3})(2-\sqrt{3})} \)
• Distributing in the numerator: \( 3(2-\sqrt{3}) = 6-3\sqrt{3} \)
• Simplifying the denominator using difference of squares: \[ (2+\sqrt{3})(2-\sqrt{3}) = 4-3 = 1 \]
• Combining the results: \[ \frac{6-3\sqrt{3}}{1} = 6-3\sqrt{3} \]

Thus, the final simplified expression is \( 6-3\sqrt{3} \).