Rationalizing Denominators: Simplify (3+√3)/(4+2√3)
Have you ever stumbled upon a fraction with a radical in the denominator and felt a bit perplexed? Don't worry; you're not alone! Rationalizing the denominator is a common technique in mathematics to eliminate radicals from the denominator of a fraction. It makes the expression easier to work with and is often required for simplifying expressions. In this comprehensive guide, we'll break down the process of rationalizing the denominator and simplifying the expression (3+√3)/(4+2√3) step by step. By the end of this article, you'll not only understand the mechanics but also grasp the underlying principles, empowering you to tackle similar problems with confidence. So, let's dive in and unravel the mystery behind rationalizing denominators!
Understanding the Basics of Rationalizing the Denominator
Before we jump into the specifics of our example, let's establish a solid foundation by understanding the core concept of rationalizing the denominator. At its heart, rationalizing the denominator means transforming a fraction so that the denominator no longer contains any radical expressions, such as square roots or cube roots. Why do we do this? Well, having a radical in the denominator can make further calculations and comparisons more difficult. By eliminating the radical, we often simplify the expression and make it easier to manipulate.
The key to rationalizing the denominator lies in a clever mathematical trick: multiplying both the numerator and the denominator of the fraction by a carefully chosen expression. This expression is usually the conjugate of the denominator. But what exactly is a conjugate? For a binomial expression like a + b√c, its conjugate is a - b√c. The conjugate has the same terms but with the opposite sign between them. When we multiply an expression by its conjugate, we eliminate the radical because of the difference of squares pattern: (a + b)(a - b) = a² - b². This eliminates the square root. In our case, the denominator is 4 + 2√3, so its conjugate will play a crucial role in simplifying the expression. Keep this concept of conjugates in mind as we move forward; it's the cornerstone of rationalizing denominators. Understanding this principle will not only help you with this particular problem but also equip you with a versatile tool for simplifying a wide range of mathematical expressions. This foundational knowledge is essential for success in algebra and beyond.
Step-by-Step Solution: Rationalizing (3+√3)/(4+2√3)
Now that we've covered the basics, let's tackle the problem at hand: simplifying the expression (3+√3)/(4+2√3). We will go through each step meticulously, so you understand not only the how but also the why behind each operation. By following along, you'll gain a deeper appreciation for the process of rationalizing denominators and simplifying complex expressions.
1. Identify the Conjugate
The first step in rationalizing the denominator is to identify the conjugate of the denominator. As we discussed earlier, the conjugate of a binomial expression a + b√c is a - b√c. In our expression, the denominator is 4 + 2√3. Applying the definition of a conjugate, we find that the conjugate of 4 + 2√3 is 4 - 2√3. This conjugate is the key to eliminating the radical in the denominator. Remember, the conjugate has the same terms as the original expression but with the sign between them flipped. Identifying the conjugate correctly is crucial, as it's the expression we'll use to multiply both the numerator and denominator.
2. Multiply Numerator and Denominator by the Conjugate
Once we've identified the conjugate, the next step is to multiply both the numerator and the denominator of the fraction by this conjugate. This is a crucial step because multiplying by the conjugate allows us to eliminate the radical in the denominator. In our case, we'll multiply both the numerator (3+√3) and the denominator (4+2√3) by the conjugate (4-2√3). This gives us the following expression:
((3+√3) * (4-2√3)) / ((4+2√3) * (4-2√3))
It's important to multiply both the numerator and the denominator by the same expression. This is equivalent to multiplying the entire fraction by 1, which doesn't change the value of the fraction, only its form. This is a common mathematical technique used to manipulate expressions without altering their fundamental value. Now that we've multiplied by the conjugate, the next step is to simplify the resulting expression. This involves carefully applying the distributive property (also known as FOIL) and combining like terms.
3. Simplify the Numerator
Now, let's simplify the numerator. We have (3+√3) * (4-2√3). To simplify this, we use the distributive property (FOIL method): First, Outer, Inner, Last. This means we multiply each term in the first binomial by each term in the second binomial:
- 3 * 4 = 12
- 3 * (-2√3) = -6√3
- √3 * 4 = 4√3
- √3 * (-2√3) = -2 * 3 = -6
Now, combine these terms: 12 - 6√3 + 4√3 - 6. We can combine the constant terms (12 and -6) and the terms with √3 (-6√3 and 4√3). This gives us:
12 - 6 = 6 -6√3 + 4√3 = -2√3
So, the simplified numerator is 6 - 2√3. This step is crucial because it sets the stage for further simplification of the entire expression. By carefully applying the distributive property and combining like terms, we've transformed the numerator into a more manageable form. Next, we'll focus on simplifying the denominator, which is where the magic of the conjugate truly comes into play.
4. Simplify the Denominator
Next, let's simplify the denominator: (4+2√3) * (4-2√3). Notice that this is in the form of (a + b)(a - b), which is a difference of squares. This pattern simplifies to a² - b². In our case, a = 4 and b = 2√3. So, we have:
4² - (2√3)²
Let's break this down:
- 4² = 16
- (2√3)² = 2² * (√3)² = 4 * 3 = 12
Now, subtract: 16 - 12 = 4. So, the simplified denominator is 4. This is a key outcome of using the conjugate. By multiplying the original denominator by its conjugate, we've successfully eliminated the radical, resulting in a simple integer. This simplification is at the heart of rationalizing the denominator. Now that we've simplified both the numerator and the denominator, we can put them together to see the simplified fraction.
5. Combine the Simplified Numerator and Denominator
Now that we've simplified both the numerator and the denominator, we can combine them to get the simplified fraction. The simplified numerator is 6 - 2√3, and the simplified denominator is 4. So, our fraction now looks like this:
(6 - 2√3) / 4
However, we're not quite done yet. We can simplify this fraction further by factoring out a common factor from the numerator. Notice that both 6 and -2 are divisible by 2. Let's factor out a 2 from the numerator:
2 * (3 - √3)
So, our fraction now looks like this:
(2 * (3 - √3)) / 4
6. Final Simplification
Finally, we can simplify the fraction by canceling the common factor of 2 between the numerator and the denominator. We have:
(2 * (3 - √3)) / 4
Divide both the numerator and the denominator by 2:
(3 - √3) / 2
And there you have it! The fully simplified form of the expression (3+√3)/(4+2√3) is (3 - √3) / 2. This is our final answer. We've successfully rationalized the denominator and simplified the expression by following a step-by-step process. This final simplification demonstrates the power of rationalizing the denominator. By eliminating the radical from the denominator, we've transformed a complex-looking fraction into a much simpler and easier-to-understand form. This skill is invaluable in many areas of mathematics, so mastering it will undoubtedly benefit you in your mathematical journey.
Conclusion: Mastering Rationalizing Denominators
Congratulations! You've successfully walked through the process of rationalizing the denominator and simplifying the expression (3+√3)/(4+2√3). By understanding the concept of conjugates and following a step-by-step approach, you've transformed a seemingly complex problem into a manageable one. Remember, the key to rationalizing denominators is to multiply both the numerator and the denominator by the conjugate of the denominator. This eliminates the radical in the denominator, making the expression simpler to work with. We've covered identifying the conjugate, multiplying the numerator and denominator by the conjugate, simplifying both the numerator and the denominator, combining them, and performing any final simplifications.
This skill is not just about solving this specific problem; it's a fundamental technique in algebra and calculus. Mastering it will open doors to solving more complex problems and understanding advanced mathematical concepts. Practice is key to solidifying your understanding. Try applying these steps to other expressions with radicals in the denominator. You can find plenty of practice problems online or in textbooks. The more you practice, the more confident you'll become in your ability to rationalize denominators and simplify expressions. Keep exploring, keep practicing, and you'll find that mathematics becomes less daunting and more engaging. Happy simplifying!
For further learning and practice on rationalizing denominators, you can explore resources like Khan Academy's article on rationalizing denominators.
