Simplifying Radicals Question 4 \$\sqrt{45u^8}\$

In the realm of mathematics, simplifying radical expressions is a fundamental skill, especially when dealing with variables. This article focuses on simplifying the expression $\sqrt{45u^8}$, where $u$ represents a positive real number. We'll break down the process step-by-step, ensuring clarity and understanding for learners of all levels. By the end of this guide, you'll be equipped to tackle similar problems with confidence. Mastering these techniques not only enhances your algebraic proficiency but also provides a solid foundation for more advanced mathematical concepts. Let’s dive into the world of radicals and unlock the secrets to simplifying complex expressions. Remember, practice is key, so work through each step carefully and don't hesitate to revisit sections as needed. This skill will prove invaluable as you continue your mathematical journey. Now, let's embark on this simplification adventure!

Understanding the Basics of Radicals

Before we tackle the specific problem, let's establish a solid understanding of the basics of radicals. A radical expression consists of a radical symbol ($\sqrt{}$), a radicand (the expression under the radical), and an index (the small number indicating the root, which is 2 for square roots). Simplifying radicals involves extracting any perfect square factors from the radicand. This process relies on the principle that $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$, where $a$ and $b$ are non-negative real numbers. Understanding this property is crucial for breaking down complex radicals into simpler forms. When dealing with variables, we also need to consider the exponents. For square roots, any variable with an even exponent can be simplified by dividing the exponent by 2. This is because $(x^n)^2 = x^{2n}$, and taking the square root of $x^{2n}$ gives us $x^n$. This principle allows us to handle variables raised to powers within the radical. Remember, the goal of simplification is to express the radical in its most basic form, where the radicand has no more perfect square factors. By grasping these fundamental concepts, you'll be well-prepared to simplify expressions like $\sqrt{45u^8}$ and beyond.

Step 1: Prime Factorization of the Constant

The first step in simplifying $\sqrt{45u^8}$ is to break down the constant, 45, into its prime factors. Prime factorization is the process of expressing a number as a product of its prime factors, which are numbers greater than 1 that are only divisible by 1 and themselves (e.g., 2, 3, 5, 7, etc.). For 45, we can start by dividing it by the smallest prime number, 3. We find that 45 = 3 × 15. Now, we need to factor 15, which is also divisible by 3. So, 15 = 3 × 5. Therefore, the prime factorization of 45 is 3 × 3 × 5, which can be written as $3^2 × 5$. This representation is crucial because it allows us to identify perfect square factors. In this case, $3^2$ is a perfect square since it is the square of 3. By identifying perfect squares, we can extract them from the radical, which is a key step in simplifying the expression. Prime factorization not only simplifies the process but also ensures that we have broken down the number completely, leaving no potential perfect square factors behind. This meticulous approach is essential for achieving the simplest form of the radical.

Step 2: Simplifying the Variable Term

Now, let's turn our attention to the variable term, $u^8$, within the radical $\sqrt{45u^8}$. Simplifying variable terms under a radical involves understanding the relationship between exponents and roots. In the case of a square root, we look for even exponents because they represent perfect squares. The exponent of $u$ is 8, which is an even number. To simplify $u^8$ under a square root, we divide the exponent by 2. So, $\sqrt{u^8} = u^{8/2} = u^4$. This is because $(u^4)^2 = u^8$. The variable term $u^4$ is now simplified and can be placed outside the radical. This process of dividing the exponent by the index of the radical (which is 2 for square roots) is a fundamental technique in simplifying radical expressions with variables. It's a direct application of the properties of exponents and roots, allowing us to extract variables raised to even powers from the radical. Remember, this method works because we are essentially finding the square root of $u^8$, which is the expression that, when squared, gives us $u^8$. The simplified variable term, $u^4$, plays a crucial role in the final simplified expression of the radical.

Step 3: Combining the Simplified Terms

Having simplified both the constant and the variable term, we can now combine them to express the simplified form of $\sqrt{45u^8}$. Recall that the prime factorization of 45 is $3^2 × 5$, and we simplified $\sqrt{u^8}$ to $u^4$. We can rewrite the original expression as $\sqrt{45u^8} = \sqrt{3^2 × 5 × u^8}$. Now, we can separate the perfect square factors: $\sqrt{3^2 × 5 × u^8} = \sqrt{3^2} × \sqrt{u^8} × \sqrt{5}$. Taking the square roots of $3^2$ and $u^8$, we get 3 and $u^4$, respectively. Therefore, the expression becomes $3 × u^4 × \sqrt{5}$. Combining these terms, we get the simplified expression $3u^4\sqrt{5}$. This final step brings together all the individual simplifications we've performed, resulting in a concise and simplified form of the original radical expression. The process of combining simplified terms highlights the power of breaking down a complex problem into smaller, manageable steps. By systematically simplifying each component, we arrive at a solution that is both accurate and elegant.

Step 4: The Final Simplified Expression

After carefully breaking down and simplifying the radical expression $\sqrt{45u^8}$, we have arrived at the final answer. By factoring the constant 45 into its prime factors ($3^2 × 5$) and simplifying the variable term $u^8$, we were able to extract perfect squares from under the radical. We found that $\sqrt{45u^8}$ simplifies to $3u^4\sqrt{5}$. This final expression is in its simplest form, with no remaining perfect square factors under the radical. The coefficient, $3u^4$, represents the terms that were extracted from the square root, while the radicand, 5, is the remaining factor that could not be further simplified. This process demonstrates the power of prime factorization and understanding the properties of radicals. The simplified expression $3u^4\sqrt{5}$ is not only mathematically accurate but also presents the information in a clear and concise manner. Mastering this simplification technique is a valuable skill in algebra and beyond, enabling you to tackle more complex problems with confidence and precision.

Conclusion: Mastering Radical Simplification

In conclusion, simplifying radicals like $\sqrt{45u^8}$ involves a systematic approach that combines prime factorization, understanding variable exponents, and applying the properties of square roots. We began by breaking down the constant term, 45, into its prime factors, identifying the perfect square $3^2$. We then simplified the variable term, $u^8$, by dividing the exponent by 2, resulting in $u^4$. Combining these simplifications, we arrived at the final simplified expression: $3u^4\sqrt{5}$. This process highlights the importance of breaking down complex problems into smaller, manageable steps. Each step, from prime factorization to variable simplification, contributes to the overall solution. By mastering these techniques, you gain a deeper understanding of radicals and their properties, which is essential for success in algebra and higher-level mathematics. The ability to simplify radicals efficiently and accurately is a valuable skill that empowers you to tackle a wide range of mathematical challenges. Remember, practice is key to mastering any mathematical concept, so continue to work through similar problems to solidify your understanding and build your confidence.