Solving A Complex Mathematical Expression: A Step-by-Step Guide

by ADMIN 64 views

Hey guys! Today, we're diving deep into a mathematical problem that looks a bit intimidating at first glance. We're going to break down this complex expression step-by-step, making it super easy to understand. Our mission? To solve: √(625/81) : √(25/9) + 3 1/3 : √1,(7) - √36 : √(6^4). So, grab your thinking caps, and let's get started!

Understanding the Order of Operations

Before we even touch the numbers, it’s crucial to understand the order of operations. Remember PEMDAS/BODMAS? This acronym tells us the sequence in which we should perform mathematical operations:

  • Parentheses / Brackets
  • Exponents / Orders
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)

Keeping this in mind will prevent us from making silly mistakes and ensure we arrive at the correct answer. So, always remember PEMDAS/BODMAS is your best friend in math! Applying this order is absolutely critical in solving our complex expression. Ignoring it would be like trying to build a house without a blueprint – things could quickly fall apart. We'll start with simplifying the square roots and then move on to the division, addition, and subtraction, making sure each step follows the correct order. This methodical approach is what turns a seemingly daunting problem into a series of manageable tasks. We’ll also convert any mixed numbers or repeating decimals into fractions to make calculations smoother and more accurate. By being meticulous and following the order of operations, we’ll confidently navigate through the expression and reach the correct solution. Trust me, guys, this systematic approach not only solves the problem but also boosts your problem-solving skills in the long run!

Breaking Down the Expression: Part 1

Let's tackle the first part of our expression: √(625/81) : √(25/9). The initial step involves calculating the square roots. The square root of 625 is 25, and the square root of 81 is 9. So, √(625/81) simplifies to 25/9. Similarly, the square root of 25 is 5, and the square root of 9 is 3. Therefore, √(25/9) becomes 5/3. Now we have 25/9 : 5/3. To divide fractions, we multiply by the reciprocal of the second fraction. This means 25/9 : 5/3 transforms into (25/9) * (3/5). Multiplying the numerators gives us 25 * 3 = 75, and multiplying the denominators gives us 9 * 5 = 45. So, we have 75/45. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 15. Dividing 75 by 15 gives us 5, and dividing 45 by 15 gives us 3. Thus, 75/45 simplifies to 5/3. So, the first part of our journey ends with a clean and simple 5/3. This meticulous breakdown showcases how we can conquer even the most intricate problems by taking them one step at a time. Remember, math is like building with LEGOs; each piece has its place, and when you fit them together correctly, you create something amazing!

Tackling the Second Part: Mixed Numbers and Repeating Decimals

Next up, we have 3 1/3 : √1,(7). First, let’s convert the mixed number 3 1/3 into an improper fraction. To do this, we multiply the whole number (3) by the denominator (3) and add the numerator (1), which gives us 3 * 3 + 1 = 10. We keep the same denominator, so 3 1/3 becomes 10/3. Now, let's deal with the repeating decimal √1,(7). The notation 1,(7) means 1.7777..., where the 7 repeats infinitely. To convert this repeating decimal to a fraction, let x = 1.7777.... Then, 10x = 17.7777.... Subtracting x from 10x, we get 9x = 16, so x = 16/9. Therefore, √1,(7) is the same as √(16/9). The square root of 16 is 4, and the square root of 9 is 3, so √(16/9) simplifies to 4/3. Now we have 10/3 : 4/3. To divide fractions, we multiply by the reciprocal of the second fraction, so 10/3 : 4/3 becomes (10/3) * (3/4). Multiplying the numerators gives us 10 * 3 = 30, and multiplying the denominators gives us 3 * 4 = 12. So, we have 30/12. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 6. Dividing 30 by 6 gives us 5, and dividing 12 by 6 gives us 2. Thus, 30/12 simplifies to 5/2. This part involved a bit of fraction gymnastics, didn't it? But hey, we nailed it! Converting mixed numbers and repeating decimals can seem tricky, but with a bit of practice, it becomes second nature. And remember, every step we take brings us closer to the final answer. So, let's keep going!

Conquering the Third Part: Exponents and Square Roots

Now, let’s tackle the last part of the expression: √36 : √(6^4). The square root of 36 is simply 6. Next, we need to evaluate √(6^4). The term 6^4 means 6 multiplied by itself four times, which is 6 * 6 * 6 * 6 = 1296. So, we have √(1296). The square root of 1296 is 36. Now we have 6 : 36. To divide 6 by 36, we can write it as a fraction: 6/36. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 6. Dividing 6 by 6 gives us 1, and dividing 36 by 6 gives us 6. Thus, 6/36 simplifies to 1/6. Boom! Another piece of the puzzle solved. This section beautifully illustrates how exponents and square roots interact. Understanding these fundamental concepts is key to unraveling complex mathematical expressions. We’re not just solving a problem here; we’re building a solid foundation in math. And with each part we conquer, our confidence grows. So, let's carry this momentum into the final stretch!

Putting It All Together: The Grand Finale

Now that we've simplified each part, let's put it all together. Our original expression was: √(625/81) : √(25/9) + 3 1/3 : √1,(7) - √36 : √(6^4). We found that:

  • √(625/81) : √(25/9) = 5/3
  • 3 1/3 : √1,(7) = 5/2
  • √36 : √(6^4) = 1/6

So our expression now looks like this: 5/3 + 5/2 - 1/6. To add and subtract fractions, we need a common denominator. The least common multiple of 3, 2, and 6 is 6. So, we'll convert each fraction to have a denominator of 6:

  • 5/3 = (5 * 2) / (3 * 2) = 10/6
  • 5/2 = (5 * 3) / (2 * 3) = 15/6
  • 1/6 remains 1/6

Now our expression is 10/6 + 15/6 - 1/6. Adding the first two fractions, we get 10/6 + 15/6 = 25/6. Then, subtracting 1/6, we have 25/6 - 1/6 = 24/6. Finally, we simplify 24/6 by dividing both the numerator and the denominator by 6, which gives us 24/6 = 4. And there we have it! The final answer is 4. Phew! We made it through the mathematical maze. This final step beautifully ties everything together, showcasing how each individual calculation contributes to the ultimate solution. It’s like watching the final piece of a jigsaw puzzle fall into place – pure satisfaction!

Conclusion: Math is an Adventure!

So, guys, we've successfully tackled a complex mathematical expression by breaking it down into smaller, manageable parts. Remember, the key is to follow the order of operations, simplify step-by-step, and convert fractions and decimals as needed. Math can be challenging, but it’s also incredibly rewarding. Every problem solved is a victory earned! Keep practicing, keep exploring, and never be afraid to dive into the fascinating world of mathematics. You've got this!