Simplifying Radicals: A Step-by-Step Guide

Hey guys! Let's dive into the world of simplifying radicals. You know, those square root, cube root, and, in this case, fourth root expressions that sometimes look a bit intimidating. But don't worry, we're going to break it down step-by-step to make it super easy to understand and conquer. We'll be tackling an expression: $\frac{\sqrt[4]{96 a^{17} b^5}}{\sqrt[4]{6 a b}}$. Our goal is to simplify this radical expression as much as possible, which means getting rid of the radicals (if we can), and making the numbers and variables inside the radical as small as possible. So, let's get started!

Understanding the Basics of Simplifying Radicals

Before we jump into our specific problem, let's refresh our memory on some key concepts. Simplifying radicals is all about rewriting radical expressions in a simpler form. This often involves extracting perfect roots from the radicand (the number or expression inside the radical). The index of the radical tells us what kind of root we're dealing with. In our case, it's a fourth root (index of 4). This means we're looking for factors that are raised to the fourth power. For example, the fourth root of 16 is 2, because 2 to the power of 4 is 16 (222*2 = 16).

When simplifying radicals, we often use the following properties:

• $\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$ (Product Property of Radicals): We can break down a radical of a product into the product of radicals.
• $\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$ (Quotient Property of Radicals): The radical of a quotient is the quotient of the radicals.

So, with these rules in mind, let's get back to our problem. We'll be using both of these properties to simplify our expression. We'll break down the numerator and the denominator, and then combine like terms to achieve our goal of simplifying this expression. This process involves a combination of understanding exponents, prime factorization, and some clever manipulation. But don't sweat it, we'll walk through it step-by-step to ensure you've got this! We'll be using the product and quotient properties to make the expression look a lot cleaner and easier to work with. These properties are our secret weapons in the fight against complicated radicals.

Step-by-Step Simplification of $\frac{\sqrt[4]{96 a^{17} b^5}}{\sqrt[4]{6 a b}}$

Alright, let's roll up our sleeves and start simplifying! We'll go through this step by step, so you can follow along easily. Remember, our aim is to simplify the expression by combining terms and extracting perfect fourth roots.

Step 1: Combine the Radicals

First, we can use the quotient property of radicals to combine the two radicals into one big radical. This gives us:

$\frac{\sqrt[4]{96 a^{17} b^5}}{\sqrt[4]{6 a b}} = \sqrt[4]{\frac{96 a^{17} b^5}{6 a b}}$

This looks a little cleaner already, right? We've gone from having a radical divided by a radical to a single radical containing a fraction. This is the first step towards simplifying the expression by putting everything under one roof, so to speak. This step simplifies our view of the expression, making it easier to manage the components within the radical. We will simplify the fraction inside the radical, which helps us to isolate terms and prepare for extracting perfect fourth roots. This step is like the foundation of our simplification process, setting the stage for the rest of the work.

Step 2: Simplify the Fraction Inside the Radical

Next, let's simplify the fraction inside the fourth root. We'll divide the numbers and subtract the exponents of the variables. This is where those exponent rules come into play! Divide 96 by 6 to get 16. For the variables, remember that when you divide variables with the same base, you subtract the exponents. So, $a^{17}/a = a^{17-1} = a^{16}$, and $b^5/b = b^{5-1} = b^4$. This simplifies the expression within the radical to:

$\sqrt[4]{\frac{96 a^{17} b^5}{6 a b}} = \sqrt[4]{16 a^{16} b^4}$

See how much simpler that looks? We've cleaned up the numbers and the variables inside the radical, making it much easier to work with. The fraction simplification is a crucial step to remove confusion and to isolate numbers and variables to apply the rules of radicals. This prepares the inside of the radical for the final simplification steps, where we will extract perfect roots. This is akin to cleaning the workspace before getting to the main task.

Step 3: Extract Perfect Fourth Roots

Now comes the fun part: extracting the perfect fourth roots! Remember, we're looking for numbers and variables that are raised to the fourth power. The fourth root of 16 is 2 (because $2^4 = 16$). The fourth root of $a^{16}$ is $a^4$ (because $(a^4)^4 = a^{16}$). And the fourth root of $b^4$ is $b$ (because $(b)^4 = b^4$).

So, we can rewrite our expression as:

$\sqrt[4]{16 a^{16} b^4} = 2 a^4 b$

And there you have it! We've successfully simplified the radical expression. We've taken a complex-looking expression and transformed it into a much simpler form. This is the core of simplifying radicals—finding the roots that can be extracted and presenting the answer in a most reduced form. We have reached the final stage where we transform the radical into its simplest possible form by extracting the roots. We have gone from something complicated to something concise and elegant. This final step is the culmination of all the previous steps, making the entire process worthwhile.

Final Answer and Conclusion

So, the simplified form of $\frac{\sqrt[4]{96 a^{17} b^5}}{\sqrt[4]{6 a b}}$ is $2 a^4 b$. Awesome, right? We’ve taken a messy radical expression and transformed it into something elegant and easy to understand. We used the quotient property to combine radicals, simplified the fraction, and extracted the perfect fourth roots. Remember, simplifying radicals is a fundamental skill in algebra and is essential for working with more advanced mathematical concepts. Keep practicing, and you'll become a pro at simplifying radicals in no time! Keep in mind that we're dealing with the fourth root, meaning any perfect fourth power will be extracted. The key is to break down the radicand and identify the factors that can be simplified. Mastering these steps will enhance your mathematical skills. Therefore, you are well on your way to becoming a radical-simplifying superstar. This is more than just math; it’s about understanding the underlying patterns and how different mathematical operations work together. Congratulations, you made it!