Simplifying The Square Root Of 75t^11u^3 A Step-by-Step Guide

In the realm of mathematics, simplifying radical expressions is a fundamental skill. This article delves into the process of simplifying the expression ${\sqrt{75 t^{11} u^3}}$, assuming that all variables represent positive real numbers. We'll break down the steps, explain the underlying principles, and provide a clear, comprehensive understanding of how to tackle such problems. Grasping simplifying radical expressions is pivotal for success in algebra and calculus, enabling students and enthusiasts to express mathematical statements in their most concise and understandable forms. This guide meticulously navigates the simplification process, ensuring that even beginners can follow along with ease while offering seasoned mathematicians a refreshed perspective. By mastering these techniques, one can confidently tackle more complex mathematical problems and appreciate the elegance of simplified expressions.

Understanding the Basics of Radicals

Before diving into the specifics of our expression, let's establish a solid foundation by reviewing the basics of radicals. A radical is a mathematical expression that involves a root, such as a square root, cube root, or nth root. The most common radical is the square root, denoted by the symbol ${\sqrt{}}$. The number inside the radical symbol is called the radicand. Simplifying radicals involves expressing the radicand in its simplest form, often by factoring out perfect squares (for square roots), perfect cubes (for cube roots), and so on. The square root of a number x is a value that, when multiplied by itself, equals x. For instance, the square root of 25 is 5 because 5 * 5 = 25. Similarly, the square root of 144 is 12 because 12 * 12 = 144. When dealing with more complex expressions, it's crucial to break down the radicand into its prime factors. This allows us to identify and extract perfect square factors, which is a key step in simplification. Additionally, understanding the properties of exponents is essential, as they play a significant role in simplifying radicals involving variables. This foundational knowledge sets the stage for effectively simplifying more intricate radical expressions.

Core Principles of Simplifying Radicals

The core principle of simplifying radicals lies in identifying and extracting perfect square factors from the radicand. This process is rooted in the property ${\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}}$, which allows us to separate the radical into manageable parts. For numerical radicands, we factor the number into its prime factors and look for pairs (in the case of square roots), triplets (for cube roots), or higher powers. For variable terms, we leverage the rule ${\sqrt{x^n} = x^{n/2}}$, where n is an even integer. This enables us to divide the exponent by 2 and extract the variable from the radical. For instance, ${\sqrt{x^4} = x^2}$ because ${x^4}$ is a perfect square. When simplifying, it is crucial to ensure that the remaining radicand has no perfect square factors left. This involves systematically checking for factors and extracting them until the expression is in its most simplified form. Mastering these core principles is fundamental to efficiently and accurately simplifying radicals, making complex expressions more accessible and easier to work with. The aim is to reduce the expression under the radical to its smallest possible form, ensuring clarity and precision in mathematical operations.

Step-by-Step Simplification of ${\sqrt{75 t^{11} u^3}}$

Now, let's apply these principles to simplify the given expression: ${\sqrt{75 t^{11} u^3}}$. This process involves several steps, each designed to break down the expression into its simplest components. The goal is to remove as much as possible from under the radical sign, leaving a simplified expression that is both mathematically equivalent and easier to understand. Each step will be explained in detail to ensure clarity and to provide a comprehensive guide for simplifying similar expressions.

Step 1: Factor the Radicand

The first step in simplifying this radical is to factor the radicand, which is ${75 t^{11} u^3}$. We begin by factoring the numerical part, 75. The prime factorization of 75 is ${3 \cdot 5^2}$. Next, we address the variable terms. For ${t^{11}}$, we can rewrite it as ${t^{10} \cdot t}$, where ${t^{10}}$ is a perfect square (since 10 is an even number). Similarly, for ${u^3}$, we can rewrite it as ${u^2 \cdot u}$, where ${u^2}$ is a perfect square. Thus, the factored form of the radicand is ${3 \cdot 5^2 \cdot t^{10} \cdot t \cdot u^2 \cdot u}$. Factoring the radicand in this way allows us to identify the perfect square components, which can then be extracted from under the radical. This step is crucial for simplifying complex radicals and is a fundamental part of the simplification process.

Step 2: Rewrite the Expression

Having factored the radicand, we can now rewrite the original expression as follows:

${ \sqrt{75 t^{11} u^3} = \sqrt{3 \cdot 5^2 \cdot t^{10} \cdot t \cdot u^2 \cdot u} }$

This rewriting step is crucial because it visually separates the perfect square factors from the non-perfect square factors. The rewritten expression clearly shows which parts of the radicand can be simplified and which parts will remain under the radical. This separation is key to applying the square root property effectively. By organizing the radicand in this manner, we prepare the expression for the next step, which involves extracting the square roots of the perfect square factors. The clarity gained in this step helps to avoid errors and ensures a systematic approach to simplification. This step is not just about rewriting; it’s about organizing the components for effective simplification.

Step 3: Apply the Product Rule of Radicals

Now, we apply the product rule of radicals, which states that ${\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}}$. Using this rule, we can separate the radical into multiple radicals, each containing a factor from the factored radicand:

${ \sqrt{3 \cdot 5^2 \cdot t^{10} \cdot t \cdot u^2 \cdot u} = \sqrt{3} \cdot \sqrt{5^2} \cdot \sqrt{t^{10}} \cdot \sqrt{t} \cdot \sqrt{u^2} \cdot \sqrt{u} }$

This separation allows us to focus on simplifying each radical individually. The product rule is a fundamental property of radicals and is essential for breaking down complex expressions into simpler ones. By applying this rule, we transform a single radical with multiple factors into a product of radicals, each containing a single factor. This makes it easier to identify and simplify the perfect squares. This step is a critical bridge in the simplification process, enabling us to move from a complex radical to a series of simpler radicals that can be easily evaluated.

Step 4: Simplify the Perfect Squares

In this step, we simplify the perfect squares. Recall that the square root of a number squared is the number itself (e.g., ${\sqrt{5^2} = 5}$), and the square root of a variable raised to an even power is the variable raised to half that power (e.g., ${\sqrt{t^{10}} = t^5}$). Applying these rules, we simplify the perfect square radicals:

${ \sqrt{5^2} = 5, \quad \sqrt{t^{10}} = t^5, \quad \sqrt{u^2} = u }$

The radicals ${\sqrt{3}}$, ${\sqrt{t}}$, and ${\sqrt{u}}$ cannot be simplified further because 3, t, and u do not have any perfect square factors. These will remain under the radical sign in the final simplified expression. Identifying and simplifying the perfect squares is a crucial step in reducing the expression to its simplest form. This step showcases the power of understanding the properties of radicals and how they interact with exponents. By extracting the square roots of the perfect squares, we significantly reduce the complexity of the expression.

Step 5: Combine the Simplified Terms

Finally, we combine the simplified terms. We multiply the terms that are outside the radicals (5, ${t^5}$, and u) and keep the terms that are still under the radicals (${\sqrt{3}}$, ${\sqrt{t}}$, and ${\sqrt{u}}$). The simplified expression is:

${ 5 t^5 u \sqrt{3 t u} }$

This is the simplest form of the original expression, ${\sqrt{75 t^{11} u^3}}$. Combining the terms involves bringing together the simplified parts in a cohesive manner. The terms extracted from the radical are multiplied together, and the remaining terms under the radical are combined back into a single radical. This step represents the culmination of the simplification process, resulting in a clean and concise expression. The final simplified form is not only mathematically equivalent to the original but also easier to interpret and work with. This step underscores the importance of organization and attention to detail in mathematical simplification.

Final Simplified Expression

After following these steps, the final simplified expression for ${\sqrt{75 t^{11} u^3}}$ is:

${ 5 t^5 u \sqrt{3 t u} }$

This expression is now in its simplest form, with no remaining perfect square factors under the radical. The final simplified expression encapsulates all the steps taken, showcasing the transformation from a complex radical to a more manageable form. This result highlights the importance of systematically applying the rules and properties of radicals to achieve simplification. The simplified expression is not only more elegant but also more practical for further mathematical operations. By mastering this process, one can confidently simplify similar radical expressions and enhance their overall mathematical proficiency. The final result is a testament to the power of simplification in making complex expressions more accessible and understandable.

Conclusion: Mastering Radical Simplification

In conclusion, mastering the simplification of radicals is a crucial skill in mathematics. By understanding the principles of factoring, applying the product rule of radicals, and simplifying perfect squares, we can transform complex expressions into their simplest forms. The step-by-step simplification of ${\sqrt{75 t^{11} u^3}}$ provided a clear example of this process. This skill not only enhances one's ability to solve mathematical problems but also fosters a deeper understanding of mathematical concepts. The ability to simplify radicals is essential for various areas of mathematics, including algebra, calculus, and beyond. It allows for clearer and more efficient problem-solving, and it is a testament to the beauty and order within mathematics. Continued practice and application of these principles will solidify your understanding and proficiency in simplifying radicals. The journey of mastering radical simplification is a rewarding one, leading to greater mathematical confidence and competence.