Solving Square Root Expressions A Step-by-Step Guide For $\sqrt{\frac{144 \times 169}{\sqrt{36}}}$

Introduction

In this comprehensive guide, we will meticulously walk through the process of solving the mathematical expression $\sqrt{\frac{144 \times 169}{\sqrt{36}}} $. This problem combines several fundamental mathematical operations, including square roots, multiplication, and division. By breaking down the expression into manageable steps, we will ensure a clear and easy-to-follow solution. Whether you're a student looking to enhance your math skills or someone who enjoys mathematical challenges, this guide will provide you with a solid understanding of how to approach and solve similar problems. Understanding the order of operations and the properties of square roots is crucial for tackling such expressions. Our step-by-step approach will not only provide the solution but also explain the reasoning behind each step. We will begin by simplifying the square root in the denominator and then proceed to simplify the numerator. Finally, we will combine these simplified components to arrive at the final answer. This method allows for a systematic and error-free solution. By the end of this guide, you will be equipped with the knowledge and confidence to solve similar complex mathematical expressions. Remember, practice is key to mastering mathematical skills, so we encourage you to try solving similar problems on your own after going through this guide. The ability to solve such expressions is a valuable skill in various fields, including engineering, physics, and computer science. It also enhances problem-solving abilities in everyday situations. Let's dive into the solution and unlock the secrets of this mathematical expression! Make sure to follow along with each step and understand the underlying principles. This will not only help you solve this particular problem but also strengthen your overall mathematical foundation.

Step 1: Simplify the Square Root in the Denominator

Our first step in solving the expression $\sqrt{\frac{144 \times 169}{\sqrt{36}}}$ is to simplify the square root in the denominator, which is $\sqrt{36}$. The square root of a number is a value that, when multiplied by itself, gives the original number. In this case, we are looking for a number that, when multiplied by itself, equals 36. We know that 6 multiplied by 6 equals 36 (6 × 6 = 36). Therefore, the square root of 36 is 6. By simplifying the denominator first, we make the overall expression easier to manage. This approach aligns with the order of operations, which often dictates that we simplify within parentheses or under radicals before performing other operations. Understanding square roots is fundamental in mathematics, and it's essential to be comfortable with finding the square roots of common numbers. This skill is not only useful in simplifying expressions but also in various other mathematical contexts, including geometry and algebra. Replacing $\sqrt{36}$ with 6, our expression now becomes $\sqrt{\frac{144 \times 169}{6}}$. This simplification sets the stage for further steps, where we will focus on simplifying the numerator and then the entire fraction under the square root. Simplifying the denominator is a strategic move because it reduces the complexity of the fraction, making subsequent calculations more straightforward. By breaking down the problem into smaller, manageable steps, we minimize the chances of errors and ensure a clear and accurate solution. Now that we have simplified the denominator, we can move on to simplifying the numerator, which involves multiplying 144 and 169. This will be the next key step in solving the expression.

Step 2: Simplify the Numerator

Having simplified the denominator to 6, we now turn our attention to the numerator of the expression $\sqrt{\frac{144 \times 169}{6}}$, which is 144 × 169. To simplify the numerator, we need to multiply these two numbers. Multiplication is a fundamental arithmetic operation, and accurately performing it is crucial for solving this expression. We can perform this multiplication manually or use a calculator for convenience. When we multiply 144 by 169, we get 24336. Therefore, the numerator simplifies to 24336. This step significantly simplifies the expression under the square root, making it easier to handle in the subsequent steps. Accurate multiplication is essential in mathematics, and it's a skill that is frequently used in various calculations and problem-solving scenarios. Understanding the process of multiplication and being able to perform it efficiently is a valuable asset in mathematics. After multiplying the numbers, our expression now looks like $\sqrt{\frac{24336}{6}}$. We have successfully simplified both the denominator and the numerator, bringing us closer to the final solution. The next step involves dividing the simplified numerator by the simplified denominator. This division will further simplify the expression under the square root, making it easier to find the final answer. By breaking down the problem into smaller, manageable steps, we have made the process more straightforward and less prone to errors. This step-by-step approach is a key strategy in solving complex mathematical problems. Now that we have the simplified numerator and denominator, we can proceed to the next step, which is dividing 24336 by 6.

Step 3: Divide the Numerator by the Denominator

With the numerator simplified to 24336 and the denominator to 6, our expression now stands as $\sqrt{\frac{24336}{6}}$. The next logical step is to perform the division of the numerator by the denominator. This operation will further simplify the expression under the square root, bringing us closer to our final answer. To divide 24336 by 6, we can use long division or a calculator. When we perform this division, we find that 24336 ÷ 6 = 4056. Therefore, the fraction under the square root simplifies to 4056. This simplification is a crucial step in solving the problem as it reduces the complexity of the expression we need to evaluate the square root of. Division is another fundamental arithmetic operation that is essential in mathematics. Being proficient in division is crucial for simplifying fractions and solving various mathematical problems. After performing the division, our expression now looks like $\sqrt{4056}$. We have successfully simplified the fraction under the square root, and now we are ready to find the square root of the resulting number. The next step involves finding the square root of 4056. This will give us the final answer to the problem. By following a step-by-step approach, we have broken down the complex expression into smaller, manageable parts, making the solution process clearer and more straightforward. Now, let's proceed to the final step, which is finding the square root of 4056.

Step 4: Calculate the Square Root

Having simplified the expression to $\sqrt{4056}$, our final step is to calculate the square root of 4056. The square root of a number is a value that, when multiplied by itself, equals the original number. In this case, we are looking for a number that, when multiplied by itself, equals 4056. Finding the exact square root of 4056 can be challenging without a calculator, as it is not a perfect square. However, we can use a calculator or estimation techniques to find an approximate value. Using a calculator, we find that the square root of 4056 is approximately 63.686733. Therefore, the solution to the expression $\sqrt{\frac{144 \times 169}{\sqrt{36}}}$ is approximately 63.69 when rounded to two decimal places. Understanding how to calculate square roots is a vital skill in mathematics and is used in various applications, including geometry, algebra, and calculus. While calculators can provide precise answers, it's also important to understand how to estimate square roots manually, which can be helpful in situations where a calculator is not available. After calculating the square root, we have arrived at the final solution to the problem. We have successfully simplified the expression step by step, from simplifying the denominator to multiplying the numerator and finally finding the square root. This step-by-step approach demonstrates a systematic way to solve complex mathematical problems. In summary, the solution to the expression $\sqrt{\frac{144 \times 169}{\sqrt{36}}}$ is approximately 63.69. This concludes our step-by-step guide to solving this mathematical expression.

Conclusion

In conclusion, we have successfully solved the mathematical expression $\sqrt{\frac{144 \times 169}{\sqrt{36}}}$ by breaking it down into a series of manageable steps. We began by simplifying the square root in the denominator, which was $\sqrt{36}$, resulting in 6. Next, we simplified the numerator by multiplying 144 and 169, which gave us 24336. Then, we divided the simplified numerator by the simplified denominator, resulting in 4056. Finally, we calculated the square root of 4056, which is approximately 63.69. This step-by-step approach not only provides the solution but also illustrates a systematic method for tackling complex mathematical problems. Mastering these fundamental mathematical operations—square roots, multiplication, division—is crucial for success in mathematics and related fields. By understanding the order of operations and applying them methodically, we can simplify complex expressions and arrive at accurate solutions. The ability to solve such problems is not only valuable in academic settings but also in various real-world applications, including engineering, physics, and computer science. Furthermore, this exercise highlights the importance of accuracy in calculations. Each step in the process builds upon the previous one, so any error in an earlier step can propagate through the solution. Therefore, it's essential to double-check each calculation and ensure that all steps are performed correctly. This guide serves as a valuable resource for students, educators, and anyone interested in enhancing their mathematical skills. By following this step-by-step approach, you can confidently tackle similar problems and strengthen your understanding of mathematical concepts. Remember, practice is key to mastering mathematical skills, so continue to practice and explore different types of problems to further enhance your abilities. We hope this guide has been helpful and informative in your mathematical journey. Keep exploring and keep solving!