Math Equation Solved: Step-by-Step Guide

Hey mathletes! Today, we're diving deep into a pretty gnarly equation:

$\frac{\sqrt{13+3^4+25 \cdot 2}}{\left((2 \cdot 3)^2-2 \cdot 45\right) \div 9}$

This looks intimidating, I know, but trust me, guys, with a little patience and a systematic approach, we can totally break it down. We'll tackle the numerator and the denominator separately, simplifying each part until we get to the final, elegant answer. Get ready to flex those math muscles!

Breaking Down the Numerator: A Journey into Square Roots

Alright, let's start with the top half of our fraction, the numerator. This is where we've got that intriguing square root symbol. Remember, to solve anything under a square root, we need to simplify the entire expression inside it first. So, let's focus on 13 + 3^4 + 25 ⋅ 2. The first thing that jumps out is 3^4. Now, what does that mean? It means 3 multiplied by itself four times: 3 * 3 * 3 * 3. That gives us 81. So, our expression becomes 13 + 81 + 25 ⋅ 2. Next up is 25 ⋅ 2. This is a simple multiplication, and 25 times 2 equals 50. Now, our numerator expression is 13 + 81 + 50. This is straightforward addition, guys. Adding these numbers together, we get 13 + 81 = 94, and then 94 + 50 = 144. So, the entire expression under the square root simplifies to 144. Our numerator is now √144. And what's the square root of 144? That's right, it's 12! We found this by recognizing that 12 * 12 = 144. So, the entire numerator simplifies down to a clean 12. High five! We've conquered the first hurdle, and it wasn't so bad, right? Keep this 12 in mind; it's going to be crucial for our final answer. Remember, always follow the order of operations (PEMDAS/BODMAS) – parentheses, exponents, multiplication and division (from left to right), and finally, addition and subtraction (from left to right). This methodical approach ensures we don't miss any steps and arrive at the correct intermediate values. The exponent 3^4 is a key part, and correctly calculating it as 81 is essential. Similarly, the multiplication 25 ⋅ 2 yielding 50 needs to be done before the additions. Finally, adding 13, 81, and 50 brings us to 144, and recognizing that √144 = 12 completes the numerator's simplification. This careful step-by-step process is what makes solving complex math problems manageable and even enjoyable.

Conquering the Denominator: Order of Operations in Action

Now, let's shift our focus to the denominator: ((2 ⋅ 3)^2 - 2 ⋅ 45) ÷ 9. This part has a few more layers, but we'll peel them back one by one. The first thing we need to deal with are the parentheses. Inside the outermost parentheses, we have (2 ⋅ 3)^2 - 2 ⋅ 45. Let's start with the inner parentheses: 2 ⋅ 3. That equals 6. So, the expression becomes (6)^2 - 2 ⋅ 45. Now, we have an exponent: (6)^2. This means 6 multiplied by itself, which is 36. So, we have 36 - 2 ⋅ 45. Next, we need to perform the multiplication: 2 ⋅ 45. That equals 90. Our expression is now 36 - 90. Performing the subtraction, 36 - 90 equals -54. So, the part inside the outermost parentheses simplifies to -54. Now, we have -54 ÷ 9. This is the final step for the denominator. Dividing -54 by 9 gives us -6. Therefore, the entire denominator simplifies to -6. Phew! We've navigated the complexities of the denominator, and the result is a solid -6. It's awesome how applying the order of operations step-by-step helps untangle these multi-part expressions. We tackled the innermost operation first (2 ⋅ 3), then the exponent ((2 ⋅ 3)^2), followed by the multiplication (2 ⋅ 45), and finally the subtraction (36 - 90). Each stage brought us closer to the simplified value of -54. The final division by 9 then led us to the denominator's value of -6. It’s crucial to pay attention to the signs, especially when dealing with negative numbers, as a simple mistake here could drastically alter the final outcome. The denominator is just as important as the numerator in a fraction, and getting it right ensures our overall solution is accurate. So, we've got our simplified numerator as 12 and our simplified denominator as -6. The final act is just around the corner!

The Grand Finale: Bringing It All Together

We've done the heavy lifting, guys! We simplified the numerator to 12 and the denominator to -6. Now, all that's left is to put them back together in the original fraction format and perform the final division. The equation now looks like this: 12 / -6. This is a simple division problem. 12 divided by -6 equals -2. And there you have it! The solution to our complex-looking equation is a neat and tidy -2. It's amazing what you can achieve by breaking down a problem into smaller, manageable steps and consistently applying the rules of mathematics. Remember this process the next time you encounter a daunting equation; the key is patience, accuracy, and a solid understanding of order of operations. So, the final answer to $\frac{\sqrt{13+3^4+25 \cdot 2}}{\left((2 \cdot 3)^2-2 \cdot 45\right) \div 9}$ is -2. Wasn't that satisfying? Keep practicing, and you'll be solving even tougher problems in no time! The journey from a complex expression to a simple integer like -2 highlights the power and beauty of mathematical simplification. Each step, from evaluating exponents and roots to performing basic arithmetic operations, played a vital role. The numerator's value of 12, derived from simplifying the square root of 144, and the denominator's value of -6, obtained through careful execution of operations within parentheses and subsequent division, were the essential building blocks. The final division of 12 by -6 resulting in -2 is the culmination of this systematic problem-solving approach. This kind of problem-solving is not just about finding an answer; it's about developing critical thinking skills and a methodical mindset that can be applied to countless other challenges in life. So, the next time you see a long math problem, don't be discouraged – see it as an opportunity to practice your problem-solving superpowers!