Simplifying Radicals How To Evaluate √5 × √215

Understanding the Problem

To determine the value of the expression $\sqrt{5} \cdot \sqrt{215}$, we need to simplify it using the properties of square roots. This problem falls under the category of basic algebraic simplification, focusing on the manipulation of radicals. Understanding and applying the rules of radicals is crucial for solving such expressions efficiently. The key concept here is the product rule for radicals, which states that the square root of a product is equal to the product of the square roots, and vice versa. This rule is particularly useful when dealing with expressions involving multiple square roots that can be combined or simplified.

Basic Principles of Radicals

Before we delve into the solution, it's essential to understand the basic principles of radicals. A radical, denoted by the symbol $\sqrt{}$, represents the root of a number. In the context of square roots, $\sqrt{a}$ signifies the number that, when multiplied by itself, equals 'a'. For instance, $\sqrt{9}$ is 3 because 3 * 3 = 9. One of the fundamental properties of radicals is the product rule, which states that $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$. This rule allows us to combine or separate radicals, making it easier to simplify expressions. Additionally, understanding perfect squares is helpful. Perfect squares are numbers that are the result of squaring an integer (e.g., 4, 9, 16, 25, etc.). Recognizing perfect square factors within a radical helps in simplifying the expression further. For example, $\sqrt{16}$ can be simplified directly to 4 because 16 is a perfect square. These foundational concepts are the building blocks for tackling more complex radical expressions.

Applying the Product Rule

The crucial step in simplifying the expression involves applying the product rule of radicals. The product rule, as mentioned earlier, states that $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$. In our case, we have $\sqrt{5} \cdot \sqrt{215}$. By applying the product rule, we can combine these two radicals into a single radical: $\sqrt{5 \cdot 215}$. This step simplifies the expression into a more manageable form. The next task is to multiply the numbers under the radical to obtain a single value. This multiplication is a straightforward arithmetic operation, but accuracy is key to arriving at the correct final answer. Once we have the product under the square root, we can then look for opportunities to simplify the radical further, possibly by identifying perfect square factors. This process is a standard technique in simplifying radical expressions and is a cornerstone of algebra.

Detailed Solution

Step 1: Combine the Radicals

Our first step is to combine the radicals using the product rule: $\sqrt{5} \cdot \sqrt{215} = \sqrt{5 \cdot 215}$. This application of the product rule is a fundamental move in simplifying expressions involving multiple square roots. By bringing the numbers under a single radical, we set the stage for further simplification. This technique is not only useful in this specific problem but also in a wide range of algebraic manipulations involving radicals. It allows us to treat the expression as a single entity, making subsequent steps more intuitive and manageable. The key is to remember that this rule only applies to multiplication (or division) of radicals, not addition or subtraction. Understanding this nuance is critical to avoid common errors in simplifying radical expressions.

Step 2: Multiply the Numbers

Next, we multiply the numbers under the radical: $5 \cdot 215 = 1075$. This is a simple arithmetic calculation, but it's essential to ensure accuracy. A small mistake here can lead to an incorrect final answer. The multiplication step consolidates the expression further, giving us $\sqrt{1075}$. Now, we have a single square root to deal with, which is a significant step towards simplification. This part of the process underscores the importance of basic arithmetic skills in algebraic manipulations. Without accurate calculations, even the correct application of algebraic rules may lead to a wrong result. The next step will involve analyzing the number under the radical (1075) to identify any perfect square factors.

Step 3: Simplify the Radical

Now, we simplify the radical $\sqrt{1075}$ by finding its prime factorization and identifying any perfect square factors. The prime factorization of 1075 is $5^2 \cdot 43$. We can rewrite $\sqrt{1075}$ as $\sqrt{5^2 \cdot 43}$. Using the product rule in reverse, we get $\sqrt{5^2} \cdot \sqrt{43}$. Since $\sqrt{5^2} = 5$, the expression simplifies to $5\sqrt{43}$. This step demonstrates a crucial technique in simplifying radicals: identifying and extracting perfect square factors. The process involves breaking down the number under the radical into its prime factors and then looking for factors that appear in pairs. Each pair of identical factors can be taken out of the radical as a single factor. In this case, we identified $5^2$ as a perfect square factor and successfully simplified the radical expression. This method is widely applicable in various mathematical contexts, making it a vital skill for anyone working with radicals.

Final Answer

Therefore, the final answer is $5\sqrt{43}$. This result is the simplified form of the original expression $\sqrt{5} \cdot \sqrt{215}$. We arrived at this answer by systematically applying the product rule of radicals and simplifying the resulting expression through prime factorization. This process highlights the importance of understanding and applying the fundamental principles of radicals in algebraic manipulations. The solution not only provides the answer but also illustrates a method that can be used to simplify a variety of radical expressions. The key steps involved combining the radicals, performing the multiplication, and then simplifying the resulting radical by identifying and extracting perfect square factors. This comprehensive approach ensures accuracy and efficiency in solving such problems.