Math Problems: Calculations And Simplifications Explained

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Math Problems: Calculations and Simplifications Explained

Hey guys! Let's dive into some cool math problems. We're gonna break down how to solve these expressions step-by-step, making sure everything is super clear and easy to follow. Get ready to flex those math muscles! We will be tackling expressions involving square roots, and basic arithmetic operations. The goal is to simplify these expressions as much as possible, applying the rules of radicals and order of operations. Let's get started and unravel these mathematical mysteries together. Understanding the simplification of radical expressions is essential in algebra and beyond. This involves recognizing perfect squares within the radicals and extracting them to simplify the expressions. Moreover, we will address the order of operations, which dictates the sequence in which calculations are performed (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, and Addition and Subtraction), ensuring accurate results. The correct application of these principles is key to successfully solving the problems. The following examples will illustrate these concepts in detail, providing a solid foundation for more complex mathematical problems.

Solving Expression A: (2โˆš675 - 5โˆš192) : (2โˆš5ยฒ)

Okay, let's start with the first problem, often denoted as 'a'. This expression involves square roots, multiplication, subtraction, and division. Our first step is to simplify the square roots. Remember, the goal is to find perfect square factors within each radical to simplify the expression. We'll break down the numbers inside the square roots to find those perfect squares. Simplifying radical expressions involves prime factorization to identify and extract perfect squares. For example, in โˆš675, we can break down 675 into its prime factors. This will help us find squares that can be simplified. Similarly, we apply the same method to โˆš192, looking for perfect square factors. This process is crucial to simplify the radical terms before performing any arithmetic operations. Once we have simplified the radicals, we can proceed with the other operations in the expression, such as multiplication, subtraction, and division. Keep in mind the order of operations: parentheses first, then multiplication and division, and finally, addition and subtraction. Now, let's see how this works in practice, simplifying each term one by one. The key to solving these types of problems lies in a clear understanding of square root properties and arithmetic operations. By meticulously simplifying each term, we will arrive at the final answer without any confusion. Make sure to always double-check your calculations to ensure accuracy.

Let's break down (2โˆš675 - 5โˆš192) : (2โˆš5ยฒ):

  1. Simplify โˆš675: โˆš675 = โˆš(225 * 3) = 15โˆš3
  2. Simplify โˆš192: โˆš192 = โˆš(64 * 3) = 8โˆš3
  3. Simplify 2โˆš5ยฒ: 2โˆš5ยฒ = 2 * 5 = 10
  4. Rewrite the expression: (2 * 15โˆš3 - 5 * 8โˆš3) / 10
  5. Simplify: (30โˆš3 - 40โˆš3) / 10 = (-10โˆš3) / 10 = -โˆš3

So, the answer for 'a' is -โˆš3. We systematically simplified each part, making sure to apply the order of operations correctly.

Solving Expression B: (โˆš125 - โˆš405) : โˆš5 - (โˆš1134 - โˆš504) : โˆš7

Alright, let's move on to the second problem, which is a bit more involved, often referred to as 'b'. This one also includes square roots, as well as division and subtraction. The strategy here is similar to before: simplify the square roots first, then handle the division and subtraction according to the order of operations. The key to simplifying such expressions lies in recognizing and extracting perfect squares within the radicals. We will simplify each radical term by finding perfect square factors. In this problem, we need to handle two separate parts of the expression before the final subtraction: (โˆš125 - โˆš405) / โˆš5 and (โˆš1134 - โˆš504) / โˆš7. Each part requires its own simplification steps. Remember to keep track of the signs and the order of operations to ensure accuracy. This is a great exercise to practice your skills in simplifying expressions with radicals and mastering the correct sequence of calculations. Make sure to double-check your calculations at each step to minimize errors. Also, pay attention to how to divide the entire expression by the radical outside the parentheses. This will help you manage the expression carefully and avoid any mistakes. Understanding the properties of square roots and applying the order of operations meticulously will help you solve this problem effectively. Let's see how we can tackle this expression step-by-step.

Here's how to solve (โˆš125 - โˆš405) : โˆš5 - (โˆš1134 - โˆš504) : โˆš7:

  1. Simplify โˆš125: โˆš125 = โˆš(25 * 5) = 5โˆš5
  2. Simplify โˆš405: โˆš405 = โˆš(81 * 5) = 9โˆš5
  3. Simplify โˆš1134: โˆš1134 = โˆš(9 * 126) = 3โˆš126 = 3โˆš(9 * 14) = 9โˆš14
  4. Simplify โˆš504: โˆš504 = โˆš(36 * 14) = 6โˆš14
  5. Rewrite the expression: (5โˆš5 - 9โˆš5) / โˆš5 - (9โˆš14 - 6โˆš14) / โˆš7
  6. Simplify within parentheses: (-4โˆš5) / โˆš5 - (3โˆš14) / โˆš7
  7. Simplify: -4 - (3โˆš(14/7)) = -4 - 3โˆš2

Therefore, the solution for 'b' is -4 - 3โˆš2. This expression shows us how to handle multiple radical terms and operations.

Solving Expression C: (โˆš726 + โˆš864) : โˆš6 - โˆš6 (โˆš216 - โˆš600)

Let's break down expression 'c'. This problem has a mix of square roots, addition, subtraction, multiplication, and division. First, we need to simplify all the square roots involved. Then, we apply the order of operations, starting with parentheses. The key is to simplify each radical individually before combining them. We will need to simplify โˆš726, โˆš864, โˆš216, and โˆš600. The order of operations dictates that we solve expressions inside parentheses first. After simplifying the radicals, we need to deal with the multiplication and division before subtraction. So we'll deal with (โˆš726 + โˆš864) / โˆš6 and โˆš6 * (โˆš216 - โˆš600) separately. Then, we subtract the results from each other. Be very careful with the signs and make sure to apply the distributive property correctly. This problem reinforces the importance of the order of operations and the ability to simplify complex radical expressions. Practice and attention to detail are key to solving this type of problem. Let's start with breaking down those radicals!

Here's how to solve (โˆš726 + โˆš864) : โˆš6 - โˆš6 (โˆš216 - โˆš600):

  1. Simplify โˆš726: โˆš726 = โˆš(121 * 6) = 11โˆš6
  2. Simplify โˆš864: โˆš864 = โˆš(144 * 6) = 12โˆš6
  3. Simplify โˆš216: โˆš216 = โˆš(36 * 6) = 6โˆš6
  4. Simplify โˆš600: โˆš600 = โˆš(100 * 6) = 10โˆš6
  5. Rewrite the expression: (11โˆš6 + 12โˆš6) / โˆš6 - โˆš6 (6โˆš6 - 10โˆš6)
  6. Simplify within parentheses: (23โˆš6) / โˆš6 - โˆš6 (-4โˆš6)
  7. Simplify: 23 - (-4 * 6) = 23 + 24 = 47

So, the answer to 'c' is 47. This problem demonstrates a good use of the order of operations in action.

Solving Expression D: โˆš5 (โˆš1445 - โˆš845) + (โˆš1620 - โˆš1805) : โˆš5

Finally, let's take a look at expression 'd'. This problem involves square roots, multiplication, addition, and division. As always, our initial step is to simplify the square roots as much as possible. Then, we need to apply the distributive property for the multiplication and handle the division correctly. First simplify the square roots, which include โˆš1445, โˆš845, โˆš1620, and โˆš1805. Notice how we are consistent with our methodology: simplifying each radical by identifying the perfect squares. After simplifying the square roots, we apply the distributive property to the first part of the expression: โˆš5 * (โˆš1445 - โˆš845). Following this, we simplify the second part: (โˆš1620 - โˆš1805) / โˆš5. Be extra careful when distributing and dividing to ensure accurate results. Make sure to keep track of the operations and signs at each stage. Remember, practice is critical, and the more you work on problems like these, the better you'll become! Let's get started!

Here's how to solve โˆš5 (โˆš1445 - โˆš845) + (โˆš1620 - โˆš1805) : โˆš5:

  1. Simplify โˆš1445: โˆš1445 = โˆš(289 * 5) = 17โˆš5
  2. Simplify โˆš845: โˆš845 = โˆš(169 * 5) = 13โˆš5
  3. Simplify โˆš1620: โˆš1620 = โˆš(324 * 5) = 18โˆš5
  4. Simplify โˆš1805: โˆš1805 = โˆš(361 * 5) = 19โˆš5
  5. Rewrite the expression: โˆš5 (17โˆš5 - 13โˆš5) + (18โˆš5 - 19โˆš5) / โˆš5
  6. Simplify within parentheses: โˆš5 (4โˆš5) + (-โˆš5) / โˆš5
  7. Simplify: 4 * 5 - 1 = 20 - 1 = 19

Thus, the final solution for 'd' is 19. That was awesome! We have successfully simplified each expression.

Conclusion: Mastering Radical Expressions

Congrats, guys! We've tackled all the expressions. We've shown how to simplify radical expressions by identifying perfect squares, and using the order of operations. Remember that simplifying radical expressions involves breaking down the numbers inside the square roots into their prime factors and then extracting any perfect squares. Make sure to always double-check your work to catch any mistakes. The key takeaways from these examples are the ability to recognize perfect squares, apply the order of operations, and manage the arithmetic operations effectively. Keep practicing and you will become more confident in simplifying complex expressions! You've got this!