Simplify Radicals: Step-by-Step Solution

Hey guys! Today, we're diving into the world of simplifying radical expressions. Radical expressions might seem intimidating at first, but with a few key techniques and a little practice, you'll be simplifying them like a pro in no time. In this guide, we'll break down the process step-by-step, using the expression $\frac{2+\sqrt{3}}{\sqrt{3}}-\frac{\sqrt{2}-2}{\sqrt{2}}$ as our main example. We'll cover everything from rationalizing denominators to combining like terms, ensuring you have a solid understanding of how to tackle these types of problems. So, buckle up and let's get started!

Understanding the Basics of Radicals

Before we jump into the simplification process, let's quickly review the fundamentals of radicals. A radical is simply a root, such as a square root, cube root, or any higher root. The most common type is the square root, denoted by the symbol $\sqrt{}$. The number inside the radical is called the radicand. For example, in $\sqrt{9}$, 9 is the radicand. Simplifying radicals often involves breaking down the radicand into its prime factors and looking for perfect squares (or cubes, etc., depending on the root). For instance, $\sqrt{12}$ can be simplified because 12 can be factored into 4 * 3, and 4 is a perfect square. Therefore, $\sqrt{12} = \sqrt{4 * 3} = \sqrt{4} * \sqrt{3} = 2\sqrt{3}$.

When dealing with radical expressions, it's crucial to remember the properties of radicals. One key property is the product rule: $\sqrt{a * b} = \sqrt{a} * \sqrt{b}$. This is what we used in the example above. Another important property is the quotient rule: $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$. These properties allow us to manipulate and simplify radical expressions effectively. Additionally, it's important to understand what it means for a radical expression to be in its simplest form. Generally, a radical expression is considered simplified if:

1. The radicand has no perfect square factors (other than 1).
2. There are no radicals in the denominator of a fraction.
3. The fraction inside the radical is simplified.

Keeping these rules in mind will help you navigate the simplification process more smoothly. Now that we have a solid foundation, let's move on to the first step in simplifying our main expression: rationalizing the denominators.

Step 1: Rationalizing the Denominators

The first thing we need to do when simplifying $\frac{2+\sqrt{3}}{\sqrt{3}}-\frac{\sqrt{2}-2}{\sqrt{2}}$ is to rationalize the denominators. Rationalizing the denominator means getting rid of any radicals in the denominator of a fraction. We do this because it's generally considered good practice to express fractions with a rational denominator. To rationalize a denominator that is a simple square root (like $\sqrt{3}$ or $\sqrt{2}$), we multiply both the numerator and the denominator by that square root. This doesn't change the value of the fraction because we're essentially multiplying by 1.

Let's start with the first term, $\frac{2+\sqrt{3}}{\sqrt{3}}$. To rationalize the denominator, we multiply both the numerator and the denominator by $\sqrt{3}$: $\frac2+\sqrt{3}}{\sqrt{3}} * \frac{\sqrt{3}}{\sqrt{3}} = \frac{(2+\sqrt{3})\sqrt{3}}{3}$ Now, we distribute the $\sqrt{3}$ in the numerator $\frac{2\sqrt{3 + \sqrt{3} * \sqrt{3}}{3} = \frac{2\sqrt{3} + 3}{3}$ So, the first term with a rationalized denominator is $\frac{2\sqrt{3} + 3}{3}$.

Next, let's rationalize the denominator of the second term, $\frac{\sqrt{2}-2}{\sqrt{2}}$. We multiply both the numerator and the denominator by $\sqrt{2}$: $\frac\sqrt{2}-2}{\sqrt{2}} * \frac{\sqrt{2}}{\sqrt{2}} = \frac{(\sqrt{2}-2)\sqrt{2}}{2}$ Distribute the $\sqrt{2}$ in the numerator $\frac{\sqrt{2 * \sqrt2} - 2\sqrt{2}}{2} = \frac{2 - 2\sqrt{2}}{2}$ So, the second term with a rationalized denominator is $\frac{2 - 2\sqrt{2}}{2}$. Now that we've rationalized both denominators, we can rewrite our original expression as $\frac{2\sqrt{3 + 3}{3} - \frac{2 - 2\sqrt{2}}{2}$ With rationalized denominators, we're now ready to move on to the next step: simplifying the fractions and finding a common denominator.

Step 2: Simplifying Fractions and Finding a Common Denominator

Now that we have rationalized the denominators, our expression looks like this: $\frac2\sqrt{3} + 3}{3} - \frac{2 - 2\sqrt{2}}{2}$ Before we can combine these fractions, we need to simplify the second fraction and find a common denominator. Let's start by simplifying the second fraction, $\frac{2 - 2\sqrt{2}}{2}$. Notice that both terms in the numerator have a common factor of 2, and the denominator is also 2. We can factor out the 2 from the numerator $\frac{2(1 - \sqrt{2)}2}$ Now, we can cancel out the 2 in the numerator and the denominator $\frac{2(1 - \sqrt{2)}2} = 1 - \sqrt{2}$ So, the simplified second term is $1 - \sqrt{2}$. Our expression now becomes $\frac{2\sqrt{3 + 3}3} - (1 - \sqrt{2})$ To subtract the second term, it’s helpful to rewrite it as a fraction with a denominator of 1 $\frac{2\sqrt{3 + 3}3} - \frac{1 - \sqrt{2}}{1}$ Now, to combine these fractions, we need a common denominator. The least common multiple of 3 and 1 is 3. So, we'll rewrite the second fraction with a denominator of 3 $\frac{1 - \sqrt{2}1} * \frac{3}{3} = \frac{3 - 3\sqrt{2}}{3}$ Our expression now looks like this $\frac{2\sqrt{3 + 3}{3} - \frac{3 - 3\sqrt{2}}{3}$ With a common denominator, we're ready for the next step: combining the numerators.

Step 3: Combining the Numerators

With a common denominator of 3, we can now combine the numerators of our fractions: $\frac2\sqrt{3} + 3}{3} - \frac{3 - 3\sqrt{2}}{3}$ To combine the numerators, we subtract the second numerator from the first $\frac{(2\sqrt{3 + 3) - (3 - 3\sqrt2})}{3}$ Be careful with the subtraction sign! We need to distribute it to both terms in the second numerator $\frac{2\sqrt{3 + 3 - 3 + 3\sqrt2}}{3}$ Now, we can simplify the numerator by combining like terms. We have a +3 and a -3, which cancel each other out $\frac{2\sqrt{3 + 3\sqrt2}}{3}$ So, our expression now simplifies to $\frac{2\sqrt{3 + 3\sqrt{2}}{3}$ At this point, we've done a lot of simplification. We've rationalized the denominators, found a common denominator, and combined the numerators. Now, let's take a look at our simplified expression and see if there's anything else we can do.

Step 4: Checking for Further Simplification

Our simplified expression is now: $\frac{2\sqrt{3} + 3\sqrt{2}}{3}$ We need to check if there are any further simplifications we can make. First, let's look at the terms inside the radicals. The radicands are 3 and 2, which are both prime numbers, so we can't simplify the square roots any further. Next, let's see if we can factor anything out of the numerator. The coefficients are 2 and 3, which don't have any common factors other than 1. Additionally, there's no common factor between the terms in the numerator and the denominator.

Therefore, the expression $\frac{2\sqrt{3} + 3\sqrt{2}}{3}$ is in its simplest form. We've successfully simplified the original expression by rationalizing the denominators, finding a common denominator, combining the numerators, and checking for further simplification. This process might seem lengthy, but with practice, you'll become much faster at it. Remember to always break down the problem into smaller steps and carefully apply the rules of radicals.

Conclusion

Simplifying radical expressions can be a rewarding challenge! In this guide, we've walked through the process of simplifying $\frac{2+\sqrt{3}}{\sqrt{3}}-\frac{\sqrt{2}-2}{\sqrt{2}}$, covering key techniques like rationalizing denominators, finding common denominators, and combining like terms. Remember, the key to mastering these types of problems is practice. Work through similar examples, and don't be afraid to break down complex expressions into smaller, more manageable steps. With a little effort, you'll be simplifying radical expressions with confidence. Keep practicing, and you'll become a math whiz in no time! If you have any questions or want to explore more examples, feel free to ask. Happy simplifying, guys!