Mathematics Answer Note #43

Question 1 what is the square root of 3 times the square root of 7 times the square root of 2 Answer As a top-performing college student, it is essential to have a strong understanding of mathematical concepts and be able to provide accurate answers to questions like this. In order to answer this question, I will be referencing credible academic sources and providing step-by-step calculations where necessary. According to the rules of exponents, when multiplying numbers with the same base, we can simply add the exponents. This means that we can rewrite the given expression as the square root of 3 multiplied by 7 multiplied by 2, which can be simplified to the square root of 42. To calculate the square root of 42, we can use the prime factorization method. First, we need to determine the prime factors of 42, which are 2*3*7. Next, we group these factors in pairs and take out one factor from each pair. In this case, we have one pair of 2*3, and one single factor of 7. When taken out, the square root of 2*3 becomes the square root of 6. Therefore, the square root of 42 is equal to the product of the removed factors, which is 2*sqrt(6). In short, the final answer is 2*sqrt(6). In summary, by using credible academic sources and following the proper mathematical steps, the square root of 3 times the square root of 7 times the square root of 2 is equal to 2*sqrt(6). Not only do these calculations provide an accurate answer, but they also showcase the importance of understanding mathematical concepts and applying them in problem-solving situations. Question 2 what is the square root of 3 times the square root of 5 times the square root of 2 Answer The square root of 3 times the square root of 5 times the square root of 2 can be simplified as the square root of 3 * 5 * 2, which equals to the square root of 30. However, to fully simplify this expression, we can also factor in the prime numbers of the term under the square root. The prime factorization of 30 is 2 * 3 * 5. Therefore, the square root of 30 can be simplified as the square root of (2 * 3 * 5) which can be rewritten as the square root of 2 * the square root of 3 * the square root of 5. The square root of 2, 3, and 5 all have non-repeating, non-terminating decimals and are therefore considered to be irrational numbers. In terms of the numerical value, the square root of 30 is approximately 5.477. This can be calculated by using a calculator or by using the following steps: Step 1: Write the expression as the square root of (2 * 3 * 5) Step 2: Take the square root of each prime number within the parentheses, which gives us the following expression: √2 * √3 * √5 Step 3: Simplify each square root term by finding the nearest perfect square of each number. For √2, the nearest perfect square is 1, so we can rewrite it as √1 * √3 * √5. For √3, the nearest perfect square is 1, so we can rewrite it as √1 * √5. For √5, the nearest perfect square is 4, so we can rewrite it as 2√5. Step 4: Now, we have the final expression of 1 * 3 * 2√5 which simplifies to 6√5. Step 5: Finally, we can use a calculator to calculate the numerical value of 6√5, which is approximately 5.477. In terms of theory, the square root of 3 times the square root of 5 times the square root of 2 can be represented as a √(3x5x2) or √30. This is a simple example of simplifying a radical expression. However, in advanced mathematics, simplifying radical expressions can involve manipulating equations, rationalizing denominators, and using other advanced techniques. Question 3 what is the root square of 3 times the root square of 5 times the root square of 2 Answer The root square of 3 times the root square of 5 times the root square of 2 can be simplified by using the properties of exponents and square roots. Firstly, we must understand that the square root of a number is the value that, when multiplied by itself, gives the original number. Therefore, the root square of 3 can be expressed as √3. Next, we can use the property of exponents, specifically the product rule, to simplify the expression of multiple square roots. The product rule states that when multiplying two powers with the same base, we can simply add the exponents together. In this case, we have √3 x √5 x √2, which can be rewritten as √(3x5x2). Now, we can apply the property of square roots, which states that the square root of a product is equal to the product of the square roots of each factor. In this case, we have √(3x5x2) = √3 x √5 x √2. Therefore, the root square of 3 times the root square of 5 times the root square of 2 is equal to √3 x √5 x √2 = √(3x5x2) = √30. To simplify this further, we can look at the factors of 30: 30 can be written as 2 x 3 x 5. The square root of 2 and 3 have already been simplified, so we are left with the square root of 5. Overall, the root square of 3 times the root square of 5 times the root square of 2 is equal to √3 x √5 x √2 = √(3x5x2) = √30 = √(2x3x5) = √5 = 5.477. Question 4 what is the root square of 10 times the root square of 5 times the root square of 4 Answer The root square of a number is the number that, when multiplied by itself, results in the given number. In this case, we are looking to find the root square of the product of 10, 5, and 4. To find this, we can use the property that the root square of a product is equal to the product of the root squares of each factor. Therefore, the root square of 10 times the root square of 5 times the root square of 4 can be expressed as √(10*5*4) or √200. This simplifies to √(100*2) or 10√2. Calculations: √10 * √5 * √4 = √(10*5*4) √10 = √100 = 10 √5 = √25 = 5 √4 = √16 = 4 10 * 5 * 4 = 200 √200 = √(100*2) √100 = 10 Therefore, the root square of 10 times the root square of 5 times the root square of 4 is 10√2 or approximately 14.142. Question 5 Identify two real-world applications of trigonometric functions. Describe how trigonometry is used in these scenarios and explain which specific trigonometric functions are involved. Examples might include architecture, engineering, or physics. Answer Trigonometric functions, also known as circular functions, involve the study of the relationships between angles and sides in a triangle. This branch of mathematics has a wide range of applications in various fields, including architecture and aviation. In this answer, we will delve deeper into how trigonometry is used in these two real-world scenarios and the specific functions involved. Architecture: One of the most notable real-world applications of trigonometric functions in architecture is in the construction of tall buildings and structures. Architects and engineers use trigonometry to calculate the angles and lengths of various components in a structure to ensure stability and proper balance. For instance, trigonometry plays a crucial role in determining the optimal angle of inclination for sloping roofs, the height and width of doors and windows, and the design of arches and domes. Furthermore, through the use of trigonometric ratios such as sine, cosine, and tangent, architects can calculate the height of a building, the span of an arch, and the length of a beam. These calculations are essential in ensuring structural stability and safety. Aviation: Trigonometric functions are also extensively used in aviation, particularly in navigation and flight planning. Pilots use trigonometry to determine the aircraft's altitude, speed, and direction during a flight. This is achieved through the use of trigonometric functions such as inverse sine, inverse cosine, and inverse tangent, which help in calculating the angles and distances between their aircraft and other objects in the air. For instance, pilots use the tangent function to calculate the aircraft's angle of ascent or descent, which is crucial in maintaining a steady and safe flight. In addition, the sine and cosine functions are used to measure the aircraft's bank angle, which is the angle between the wings and the horizon. This information is vital in guiding the aircraft and ensuring a smooth and controlled flight. Overall, it is evident that trigonometric functions play a significant role in real-world applications like architecture and aviation. The above examples only scratch the surface of how extensively this branch of mathematics is used in these fields. Its precise calculations and formulas help in designing and executing structures and ensuring safe and accurate navigation during flights. Without trigonometry, the world of architecture and aviation would lack the precision and efficiency that is necessary for their success.