Simplifying Radicals: $\sqrt[5]{128 S^{15} T^7}$ Explained

Hey guys! Today, we're diving into the fascinating world of radical expressions, and we're going to tackle a specific problem: simplifying $\sqrt[5]{128 s^{15} t^7}$. Don't worry if it looks intimidating at first; we'll break it down step by step. This is a crucial skill in mathematics, especially in algebra and calculus, so let's get started!

Understanding the Basics of Radicals

Before we jump into the problem, let's quickly recap what radicals are. A radical expression consists of a radical symbol ($\sqrt[n]{}$), a radicand (the expression under the radical), and an index (the small number n indicating the root). In our case, the index is 5, meaning we're dealing with a fifth root. Understanding the parts of a radical expression helps in simplifying it effectively. The goal of simplifying a radical expression is to pull out any perfect nth powers from the radicand, leaving a simplified expression under the radical. This makes the expression easier to work with and understand.

Remember, simplifying radicals is like untangling a knot. We want to pull out the perfect "pieces" that can be expressed without the radical, leaving the rest in a neater, more manageable form. This process not only simplifies the expression visually but also makes it easier to use in further calculations or algebraic manipulations. The index of the radical is key because it tells us what kind of "pieces" we're looking for – squares for square roots, cubes for cube roots, and so on. In our case, since we're dealing with a fifth root, we need to identify factors that appear five times within the radicand.

Breaking Down the Radicand: 128

Okay, so let's start with the numerical part of our radicand: 128. To simplify $\sqrt[5]{128}$, we need to find the prime factorization of 128. This means breaking it down into its prime factors, which are prime numbers that multiply together to give us 128. The prime factorization of 128 is $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$, or $2^7$. Now, since we're taking the fifth root, we want to see if we can pull out any groups of five. We can rewrite $2^7$ as $2^5 \times 2^2$. This is crucial because $2^5$ is a perfect fifth power, which means we can simplify it outside the radical. When simplifying radicals, it’s always a good idea to break down the numerical part of the radicand into its prime factors. This helps in identifying perfect powers that can be extracted. In this instance, recognizing that 128 is $2^7$ allows us to easily spot the $2^5$ factor, which is key to simplifying the fifth root.

Decomposing the number into its prime factors is a fundamental step in simplifying radicals. By doing so, we expose any perfect nth powers lurking within the number. In our case, understanding that $2^7$ contains a $2^5$ term is vital for the next steps. This method is not just useful for numerical coefficients but also for variables with exponents. We’ll see how to apply a similar approach when we deal with the $s^{15}$ and $t^7$ terms in our expression. Remember, simplifying radicals is all about finding and extracting these perfect powers, leaving a cleaner and more manageable expression behind.

Simplifying the Variable Terms: $s^{15}$ and $t^7$

Now, let's tackle the variable parts: $s^{15}$ and $t^7$. Dealing with variables inside radicals is actually quite straightforward, especially when you understand the rules of exponents. For $s^{15}$, we can directly apply the rule that $\sqrt[n]{x^m} = x^{m/n}$. So, $\sqrt[5]{s^{15}} = s^{15/5} = s^3$. Easy peasy! This is because $s^{15}$ is a perfect fifth power. When the exponent of the variable is a multiple of the index, like in this case, the simplification is clean and simple. We can extract the entire variable term from the radical without any remainder.

However, for $t^7$, it's a bit different. We can rewrite $t^7$ as $t^{5} \times t^{2}$. Similar to what we did with the number 128, we're looking for the highest multiple of the index (5) that is less than or equal to the exponent (7). We rewrite the variable term in such a way that one part is a perfect fifth power ($t^5$), and the other part ($t^2$) will remain under the radical. This method is crucial for simplifying variable terms that don't have exponents that are perfect multiples of the index. By separating the variable term into a perfect power and a remainder, we can easily extract the perfect power while leaving the remainder under the radical sign.

Putting It All Together

Alright, we've broken down each component of the radicand. Now, let's put it all back together and simplify the original expression: $\sqrt[5]{128 s^{15} t^7}$. We found that:

• $\sqrt[5]{128} = \sqrt[5]{2^5 \times 2^2} = 2\sqrt[5]{2^2} = 2\sqrt[5]{4}$
• $\sqrt[5]{s^{15}} = s^3$
• $\sqrt[5]{t^7} = \sqrt[5]{t^5 \times t^2} = t\sqrt[5]{t^2}$

Now, we multiply these simplified parts together: $2 \times s^3 \times t \times \sqrt[5]{4} \times \sqrt[5]{t^2}$. To combine the radicals, we multiply the terms inside the radicals, giving us $\sqrt[5]{4t^2}$. Therefore, the simplified expression is $2s^3t\sqrt[5]{4t^2}$. This is the final simplified form of the radical expression. Notice how we extracted all the perfect fifth powers, leaving only the necessary terms under the radical. Putting it all together like this helps showcase the beauty of simplification and how each step contributes to the final answer.

Combining the simplified components is the final act in this mathematical performance. It's where all the individual simplifications come together to form a cohesive and elegant result. By carefully combining the coefficients, variables, and remaining radicals, we ensure that the final expression is not only simplified but also accurately represents the original radical. This process highlights the importance of each step in simplification and how they collectively contribute to the ultimate solution.

Final Answer and Conclusion

So, the simplified radical form of $\sqrt[5]{128 s^{15} t^7}$ is $2s^3t\sqrt[5]{4t^2}$. Great job, guys! We've successfully navigated this radical expression. Remember, the key is to break down the problem into smaller, manageable parts, identify perfect powers, and then put it all back together. Simplifying radical expressions might seem daunting at first, but with practice and a clear understanding of the rules, you'll become a pro in no time!

This example showcases the power of simplifying radicals. By breaking down complex expressions into their simplest forms, we make them easier to understand and work with. Whether you're solving algebraic equations, tackling calculus problems, or simply exploring the beauty of mathematics, simplifying radicals is a valuable skill to have. So keep practicing, keep exploring, and keep simplifying!