Simplifying Radicals: How To Simplify √2 / (3 + √8)

Hey guys, ever stumbled upon a radical expression that looks like it belongs more in a textbook than in your everyday life? Well, today, we're diving deep into simplifying one such expression: $\frac{\sqrt{2}}{3+\sqrt{8}}$. Don't worry, it's not as scary as it looks! We'll break it down step-by-step so you can conquer similar problems with confidence. Let's get started and turn this radical riddle into a piece of cake!

Understanding the Basics

Before we jump into the main problem, let's brush up on some fundamental concepts. Radicals, at their core, are ways of representing roots of numbers. The most common radical is the square root, denoted by $\sqrt{}$. For example, $\sqrt{9} = 3$ because 3 multiplied by itself equals 9. Understanding this basic concept is super important for simplifying more complex radical expressions. Think of radicals as a way to undo exponents, specifically fractional exponents. So, $\sqrt{x}$ is the same as $x^{\frac{1}{2}}$. This connection becomes incredibly useful when applying exponent rules to simplify expressions. Now, when dealing with radicals, there are a few key rules we need to remember. First, $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$, which means we can split the square root of a product into the product of square roots. Second, $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$, allowing us to separate the square root of a quotient into a quotient of square roots. And third, we always aim to remove any perfect square factors from inside the radical. For example, $\sqrt{8}$ can be simplified to $\sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}$. Mastering these rules and understanding the fundamental concepts is key to tackling more complex problems. So, keep practicing and you'll become a radical simplification pro in no time!

Simplifying the Denominator

The expression we're tackling is $\frac{\sqrt{2}}{3+\sqrt{8}}$. The first thing we want to do is simplify the denominator, specifically the $\sqrt{8}$ term. We know that 8 can be written as $4 \times 2$, and since 4 is a perfect square, we can simplify $\sqrt{8}$ as follows: $\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$. So, our expression now looks like this: $\frac{\sqrt{2}}{3+2\sqrt{2}}$. The next step involves a technique called rationalizing the denominator. This means we want to get rid of the radical in the denominator. To do this, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $3+2\sqrt{2}$ is $3-2\sqrt{2}$. So, we multiply both the numerator and the denominator by $3-2\sqrt{2}$: $\frac{\sqrt{2}}{3+2\sqrt{2}} \times \frac{3-2\sqrt{2}}{3-2\sqrt{2}}$. This might seem a bit complicated, but it's a standard trick to eliminate radicals from the denominator. This process will help us transform the expression into a simpler form where there are no radicals in the denominator, making it easier to work with and understand. Keep following along, and you'll see how this process simplifies everything neatly!

Rationalizing the Denominator

Now that we've set up the multiplication, let's perform it. We have $\frac{\sqrt{2}(3-2\sqrt{2})}{(3+2\sqrt{2})(3-2\sqrt{2})}$. First, let's expand the numerator: $\sqrt{2}(3-2\sqrt{2}) = 3\sqrt{2} - 2(\sqrt{2})^2 = 3\sqrt{2} - 2(2) = 3\sqrt{2} - 4$. Next, let's expand the denominator. Notice that the denominator is in the form of $(a+b)(a-b)$, which simplifies to $a^2 - b^2$. So, $(3+2\sqrt{2})(3-2\sqrt{2}) = 3^2 - (2\sqrt{2})^2 = 9 - 4(2) = 9 - 8 = 1$. Therefore, our expression becomes $\frac{3\sqrt{2} - 4}{1}$, which simplifies to $3\sqrt{2} - 4$. So, the simplified form of the original expression is $3\sqrt{2} - 4$. This means that $\frac{\sqrt{2}}{3+\sqrt{8}}$ is equivalent to $3\sqrt{2} - 4$. The key here was to recognize the structure of the denominator and use its conjugate to rationalize it, eliminating the radical and making the expression much cleaner and easier to understand. This technique is a cornerstone in simplifying radical expressions, and mastering it will greatly enhance your algebra skills!

Final Answer

Alright, after walking through all the steps, we've arrived at the simplified form of the expression $\frac{\sqrt{2}}{3+\sqrt{8}}$. We started by simplifying the radical in the denominator, turning $\sqrt{8}$ into $2\sqrt{2}$. Then, we rationalized the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator, which was $3-2\sqrt{2}$. After performing the multiplication and simplifying, we found that the expression simplifies to $3\sqrt{2} - 4$. So, the final answer is: $3\sqrt{2} - 4$. This entire process highlights the importance of understanding radical properties and knowing how to manipulate expressions to eliminate radicals from the denominator. By using the conjugate, we were able to transform a complex-looking fraction into a much simpler and more manageable expression. Remember, practice makes perfect, so keep working on similar problems to build your confidence and skills in simplifying radicals. You've got this!