Simplifying $5√3 ⋅ 2√6$: A Step-by-Step Guide

Are you looking to simplify the expression $5√3 ⋅ 2√6$? This comprehensive guide will walk you through each step, ensuring you understand the process thoroughly. We'll break down the expression, apply the rules of radicals, and arrive at the simplest form. Whether you're a student tackling algebra or just brushing up on your math skills, this article will provide a clear and concise explanation.

Understanding the Basics of Radicals

Before we dive into simplifying $5√3 ⋅ 2√6$, it's crucial to understand the fundamentals of radicals. Radicals, often represented by the square root symbol (√), indicate the root of a number. For example, √9 represents the square root of 9, which is 3 because 3 * 3 = 9. Similarly, √16 is 4 because 4 * 4 = 16. When dealing with expressions involving radicals, we need to know how to multiply them and simplify the results.

The key principle we'll use is the product rule for radicals, which states that √(a * b) = √a * √b. This rule allows us to combine and separate radicals, making simplification easier. Understanding this rule is fundamental to mastering radical expressions. Radicals are an essential part of algebra and calculus, so grasping these concepts is very important for mathematical proficiency. In our case, we will use this principle to first multiply the radicals and then simplify the resulting radical by finding perfect square factors.

Furthermore, it's important to remember that we can multiply the coefficients (the numbers outside the radical) separately from the radicals themselves. For instance, in the expression 2√5, 2 is the coefficient and √5 is the radical. When multiplying expressions like this, we multiply the coefficients together and the radicals together. This understanding sets the stage for simplifying more complex expressions involving radicals, and it is the foundation for our step-by-step simplification of $5√3 ⋅ 2√6$.

Step 1: Multiplying the Coefficients and Radicals

To begin simplifying the expression $5√3 ⋅ 2√6$, we first multiply the coefficients and then multiply the radicals. The coefficients are the numbers outside the square root symbols, which are 5 and 2 in this case. The radicals are √3 and √6. Multiplying the coefficients is straightforward: 5 * 2 = 10. Next, we multiply the radicals: √3 * √6. According to the product rule for radicals (√(a * b) = √a * √b), we can combine these under a single square root: √3 * √6 = √(3 * 6) = √18.

Now, we combine the results of multiplying the coefficients and the radicals. We have 10 from the coefficients and √18 from the radicals. Thus, our expression becomes 10√18. This step is crucial because it consolidates the original expression into a form that is easier to simplify further. It's important to ensure accuracy in this step, as any errors here will propagate through the rest of the simplification process. We are now looking at an expression with a single radical, which makes it easier to identify potential simplifications by finding perfect square factors within the radical.

This step highlights the importance of breaking down complex problems into simpler parts. By addressing the coefficients and radicals separately, we make the multiplication process more manageable. This approach is a common strategy in mathematics, where complex problems are often solved by breaking them down into smaller, more manageable steps. So, we've successfully multiplied the coefficients and radicals, setting the stage for the next step: simplifying the radical √18. Stay tuned as we continue to break down and conquer this mathematical problem.

Step 2: Simplifying the Radical √18

The next crucial step in simplifying $5√3 ⋅ 2√6$ is to simplify the radical √18. To simplify a radical, we look for perfect square factors within the radicand (the number inside the square root). A perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25, etc.). In the case of 18, we can factor it as 9 * 2. Notice that 9 is a perfect square (3 * 3 = 9).

We can rewrite √18 as √(9 * 2). Now, using the product rule for radicals in reverse (√a * √b = √(a * b)), we can separate this into √9 * √2. Since √9 is 3, we have 3√2. This simplification is significant because it reduces the radical to its simplest form. The factor 2 inside the square root cannot be further simplified as it has no perfect square factors other than 1. So, 3√2 is the simplified form of √18. Understanding how to identify and extract perfect square factors is crucial for simplifying radicals efficiently. This process not only simplifies the expression but also makes it easier to work with in further calculations.

This technique of breaking down the radicand into its factors and identifying perfect squares is a fundamental skill in algebra. It allows us to express radicals in their simplest form, which is essential for accurate calculations and comparisons. Moreover, this skill is not limited to square roots; it can be extended to cube roots, fourth roots, and other higher-order radicals. Therefore, mastering this step will significantly enhance your ability to manipulate and simplify various types of radical expressions. With √18 simplified to 3√2, we're now ready to substitute this back into our expression and complete the simplification process.

Step 3: Substituting Back and Final Simplification

Now that we've simplified √18 to 3√2, we can substitute this back into our expression from Step 1, which was 10√18. Replacing √18 with 3√2, we get 10 * (3√2). This step brings us closer to the final simplified form. The expression now looks much cleaner and easier to handle. It's a testament to the power of breaking down a problem into manageable parts and simplifying each part before putting them back together.

Next, we simply multiply the coefficient 10 by the coefficient 3 in 3√2. This gives us 10 * 3√2 = 30√2. And that's it! We've arrived at the final simplified form of the expression. The expression 30√2 is the simplest form because the radical √2 cannot be simplified further, and there are no remaining common factors between the coefficient and the radicand. This final step demonstrates the importance of combining the simplified parts back into the original expression to achieve the overall simplification. It’s a culmination of the previous steps, bringing us to a clear and concise answer.

Therefore, the simplified form of $5√3 ⋅ 2√6$ is 30√2. This journey through simplifying the expression highlights several key mathematical principles, including the product rule for radicals and the importance of identifying perfect square factors. Mastering these concepts not only allows you to simplify radical expressions but also builds a solid foundation for more advanced topics in mathematics. Remember, practice is key to proficiency, so keep simplifying radical expressions to strengthen your skills.

Conclusion

In conclusion, simplifying the expression $5√3 ⋅ 2√6$ involves multiplying coefficients and radicals, simplifying radicals by finding perfect square factors, and then combining the results. By following these steps, we've shown that $5√3 ⋅ 2√6$ simplifies to 30√2. This process not only provides the final answer but also enhances our understanding of radical simplification, which is a valuable skill in algebra and beyond. Remember to practice these steps with various expressions to build confidence and mastery.

Understanding the fundamentals of radicals and their simplification is crucial for success in mathematics. By breaking down complex expressions into smaller, more manageable parts, we can tackle even the most challenging problems. Always remember the product rule for radicals, how to identify perfect square factors, and the importance of careful calculation. With these tools, you'll be well-equipped to simplify radical expressions with ease. For further learning and practice on radical expressions, you can visit resources like Khan Academy.