How To Calculate The Value Of √45 × √20

In the realm of mathematics, simplifying expressions involving square roots is a fundamental skill. This article delves into the process of finding the value of √45 × √20, providing a step-by-step guide that not only arrives at the solution but also elucidates the underlying principles. Whether you are a student grappling with algebra or simply someone seeking to refresh your mathematical prowess, this exploration will equip you with the knowledge and techniques to confidently tackle similar problems.

Understanding Square Roots: The Foundation

Before we dive into the specifics of finding the value of √45 × √20, let's first establish a firm understanding of what square roots represent. The square root of a number is a value that, when multiplied by itself, equals the original number. For instance, the square root of 9 is 3 because 3 × 3 = 9. Symbolically, the square root is denoted by the radical symbol '√'.

Square roots can be whole numbers (like √9 = 3), but they can also be irrational numbers, meaning their decimal representation goes on infinitely without repeating (like √2 ≈ 1.414). Understanding this distinction is crucial when simplifying expressions, as we often aim to express square roots in their simplest form. This usually involves factoring out perfect squares from the radicand (the number under the radical symbol).

When dealing with the product of square roots, a key property comes into play: √(a × b) = √a × √b. This property allows us to break down complex square roots into simpler components, making them easier to manage and ultimately simplify. This principle will be central to our approach in evaluating √45 × √20. By understanding the fundamental concept of square roots and their properties, we lay the groundwork for confidently tackling more intricate mathematical challenges.

Deconstructing √45: Simplifying the First Square Root

Our journey to determine the value of √45 × √20 begins with simplifying each square root individually. Let's focus on √45 first. The key to simplifying square roots lies in identifying perfect square factors within the radicand. A perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25, etc.).

To simplify √45, we need to find the largest perfect square that divides 45. We can start by listing the factors of 45: 1, 3, 5, 9, 15, and 45. Among these factors, 9 stands out as a perfect square (since 3 × 3 = 9). We can therefore rewrite 45 as 9 × 5.

Now, we can express √45 as √(9 × 5). Using the property √(a × b) = √a × √b, we can separate this into √9 × √5. We know that √9 = 3, so we have 3 × √5, or simply 3√5. This is the simplified form of √45. By breaking down the radicand into its perfect square factors, we've transformed the original square root into a more manageable expression. This step is crucial for effectively calculating the value of the overall expression √45 × √20. Understanding how to simplify individual square roots is a cornerstone of working with radicals in mathematics.

Deconstructing √20: Simplifying the Second Square Root

Having successfully simplified √45, our next step in finding the value of √45 × √20 is to simplify √20. We'll employ the same strategy of identifying perfect square factors within the radicand, 20.

Let's list the factors of 20: 1, 2, 4, 5, 10, and 20. Among these factors, 4 is a perfect square (since 2 × 2 = 4). We can thus rewrite 20 as 4 × 5.

Now, we can express √20 as √(4 × 5). Applying the property √(a × b) = √a × √b, we separate this into √4 × √5. We know that √4 = 2, resulting in 2 × √5, or simply 2√5. This is the simplified form of √20. Just as we did with √45, breaking down √20 into its simplest form makes it easier to work with in further calculations. The ability to identify and extract perfect square factors is a fundamental skill in simplifying radical expressions, paving the way for accurate and efficient problem-solving in mathematics.

Multiplying Simplified Square Roots: Putting it All Together

With both √45 and √20 simplified, we can now proceed to calculate the value of √45 × √20. We've established that √45 = 3√5 and √20 = 2√5. Therefore, the original expression can be rewritten as (3√5) × (2√5).

To multiply these expressions, we multiply the coefficients (the numbers outside the square roots) and the radicands (the numbers inside the square roots) separately. This follows the principle that (a√b) × (c√d) = (a × c)√(b × d).

In our case, we multiply 3 and 2 to get 6, and we multiply √5 and √5 to get √(5 × 5) = √25. So, we have 6 × √25. Since √25 = 5, the expression further simplifies to 6 × 5. Finally, multiplying 6 and 5 gives us 30.

Therefore, √45 × √20 = 30. This result demonstrates how simplifying square roots before multiplying can significantly streamline the calculation process. By breaking down the problem into smaller, manageable steps, we were able to arrive at the solution with clarity and precision. This approach highlights the power of understanding the underlying principles of mathematics in tackling complex problems.

The Final Answer: √45 × √20 = 30

In conclusion, we have successfully navigated the process of finding the value of √45 × √20. By systematically simplifying each square root individually and then applying the rules of multiplication, we arrived at the definitive answer: 30. This journey has underscored the importance of understanding the properties of square roots, particularly the ability to factor out perfect squares and the rule for multiplying radicals.

Our step-by-step approach not only provided the solution but also illuminated the underlying mathematical principles at play. This comprehensive understanding is crucial for tackling similar problems with confidence and accuracy. Whether you're a student honing your algebra skills or an individual seeking to expand your mathematical knowledge, the techniques presented here will serve as a valuable tool in your problem-solving arsenal.

The ability to simplify and manipulate square roots is a fundamental skill in mathematics, with applications extending far beyond this specific example. By mastering these techniques, you'll be well-equipped to tackle a wide range of algebraic and mathematical challenges. Remember, the key to success lies in understanding the underlying principles and applying them systematically.

In summary, the solution to √45 × √20 is 30. This result is achieved by first simplifying √45 to 3√5 and √20 to 2√5, then multiplying the simplified forms together. The ability to simplify square roots and apply the rules of multiplication are essential skills in mathematics.