Simplifying The Expression: 7√100 - 3√36

Understanding the Basics of Square Roots

When we talk about simplifying mathematical expressions involving square roots, it's essential to first understand what a square root is. A square root of a number x is a value that, when multiplied by itself, gives x. For example, the square root of 100 is 10 because 10 * 10 = 100. Similarly, the square root of 36 is 6 because 6 * 6 = 36. Understanding this basic concept is crucial for solving the expression $7 \sqrt{100} - 3 \sqrt{36}$. This expression combines square roots with multiplication and subtraction, making it a practical example of how these mathematical operations interact. Before diving into the step-by-step simplification, let's highlight the importance of knowing perfect squares. Perfect squares are numbers that have whole number square roots, like 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. Recognizing these numbers allows for quicker simplification of square root expressions. In our case, 100 and 36 are perfect squares, which makes the simplification process more straightforward. Moreover, understanding the properties of square roots, such as $\sqrt{a * b} = \sqrt{a} * \sqrt{b}$, can further aid in simplifying more complex expressions. These foundational concepts set the stage for tackling the given problem and other similar mathematical challenges.

Step-by-Step Solution of 7√100 - 3√36

To solve the expression $7 \sqrt{100} - 3 \sqrt{36}$, we need to break it down into manageable steps. Firstly, we address the square roots. We know that $\sqrt{100}$ is 10 and $\sqrt{36}$ is 6. So, we can replace these square roots with their numerical values in the expression. This gives us $7 * 10 - 3 * 6$. Secondly, we perform the multiplications. 7 multiplied by 10 is 70, and 3 multiplied by 6 is 18. Now our expression looks like this: $70 - 18$. Finally, we subtract 18 from 70. This simple subtraction leads us to the result: $70 - 18 = 52$. Therefore, the simplified value of the expression $7 \sqrt{100} - 3 \sqrt{36}$ is 52. This step-by-step approach highlights the importance of following the order of operations (PEMDAS/BODMAS) in mathematics: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). By first dealing with the square roots, then the multiplications, and finally the subtraction, we ensure an accurate result. This methodical approach is crucial for tackling more complex mathematical problems as well. In summary, understanding each step not only helps in solving the current problem but also builds a solid foundation for future mathematical endeavors.

Detailed Breakdown of Each Operation

Let's delve deeper into each operation within the expression $7 \sqrt{100} - 3 \sqrt{36}$ to ensure a comprehensive understanding. First, consider the square root operation. The square root of a number x is a value y such that y * y = x. In our expression, we have $\sqrt{100}$ and $\sqrt{36}$. As previously mentioned, $\sqrt{100} = 10$ because 10 * 10 = 100, and $\sqrt{36} = 6$ because 6 * 6 = 36. This operation is fundamental and transforms the expression from involving radicals to simple multiplication and subtraction. Second, we tackle the multiplication operations. After evaluating the square roots, we are left with $7 * 10$ and $3 * 6$. Multiplication is a basic arithmetic operation where we are essentially scaling one number by another. 7 multiplied by 10 results in 70, and 3 multiplied by 6 results in 18. These multiplications are straightforward but necessary to simplify the expression further. Finally, we perform the subtraction. Subtraction is the arithmetic operation that represents the removal of objects from a collection. In our case, we are subtracting 18 from 70. The result of this subtraction, $70 - 18$, is 52. This final subtraction gives us the simplified value of the entire expression. By breaking down each operation—square root, multiplication, and subtraction—we not only understand the mathematical process but also appreciate the interconnectedness of these operations. This detailed analysis is essential for building confidence in solving more intricate mathematical problems. This thorough approach underscores the importance of mastering basic arithmetic principles for tackling advanced mathematical concepts.

Alternative Methods for Simplifying Expressions

While the step-by-step method described above is a direct approach to simplifying $7 \sqrt{100} - 3 \sqrt{36}$, it's always beneficial to explore alternative methods to enhance understanding and problem-solving skills. One alternative approach involves recognizing common factors, although in this specific expression, that's not directly applicable. However, for more complex expressions, factoring can significantly simplify the process. Another method, particularly useful for expressions involving variables, is to simplify the expression algebraically before substituting numerical values. In our case, since we have constants, this approach doesn't offer a significant advantage, but it's a valuable technique for algebraic expressions. For instance, if we had an expression like $a \sqrt{100} - b \sqrt{36}$, we would still simplify the square roots to get $10a - 6b$, and further simplification would depend on the values or relationship between a and b. Furthermore, understanding the properties of square roots, such as $\sqrt{a^2} = |a|$, is crucial for dealing with variables and negative numbers. While this property isn't directly used in our problem, it's essential for a comprehensive understanding of square roots. Additionally, for more complex expressions involving multiple terms and operations, breaking the expression into smaller, manageable parts can help in avoiding errors and ensuring clarity. This method involves tackling one part of the expression at a time, simplifying it, and then moving on to the next part. In conclusion, while the direct step-by-step method works effectively for $7 \sqrt{100} - 3 \sqrt{36}$, exploring alternative methods broadens mathematical skills and prepares one for more challenging problems. These alternative approaches emphasize the importance of flexibility and adaptability in problem-solving.

Real-World Applications of Square Root Simplification

The simplification of expressions involving square roots, like the one we've discussed, might seem purely academic, but it has numerous real-world applications across various fields. In physics, square roots are frequently encountered in calculations involving motion, energy, and waves. For example, the speed of an object falling under gravity involves a square root, and simplifying such expressions can help in determining the final velocity or the time taken for the object to fall. In engineering, particularly in structural and civil engineering, square roots are used in calculations related to stress, strain, and the stability of structures. Simplifying these expressions ensures accurate designs and safe constructions. In computer graphics and game development, square roots are essential for calculating distances and transformations. For instance, the distance between two points in a 2D or 3D space is calculated using the distance formula, which involves square roots. Simplifying these expressions improves the efficiency of rendering and animation processes. In finance, square roots are used in calculating standard deviations, which measure the volatility of investments. Simplifying these calculations helps in assessing risk and making informed investment decisions. In everyday life, square roots can be used in practical situations such as calculating the diagonal of a rectangular room or determining the size of a square garden. Understanding how to simplify these expressions allows for quicker and more accurate measurements. These examples illustrate that the seemingly abstract concept of simplifying square root expressions has tangible applications in a wide range of disciplines. By mastering these mathematical skills, individuals can better understand and interact with the world around them. The ability to simplify such expressions underscores the practical importance of mathematical literacy.

In conclusion, the simplified value of the expression $7 \sqrt{100} - 3 \sqrt{36}$ is 52. We arrived at this answer by first evaluating the square roots, then performing the multiplications, and finally, the subtraction. This step-by-step process highlights the importance of understanding basic mathematical operations and their order. For further learning on similar topics, you may find resources on websites like Khan Academy very helpful.