Solving 5⁷ × 3 ⁹⁄₁₂ + ⁹√49 A Step-by-Step Guide

Hey guys! Ever stumbled upon a math problem that looks like it's speaking another language? Well, today we're going to tackle one of those beasts and break it down step-by-step. We've got this equation: 5⁷ × 3 ⁹⁄₁₂ + ⁹√49, and trust me, it's not as scary as it looks. We'll go through each part, making sure you understand not just the answer, but why it's the answer. So, grab your calculators (or your brainpower!) and let's dive in!

Decoding the Equation: A Step-by-Step Guide

So, our main goal here is to simplify and solve 5⁷ × 3 ⁹⁄₁₂ + ⁹√49. To do this effectively, we're going to break it down into manageable chunks. Think of it like eating an elephant – you gotta do it one bite at a time! We'll start with the exponents, then move onto the mixed fraction, the radical, and finally, we'll put it all together. Ready? Let’s go!

Part 1: Cracking the Exponent – 5⁷

Okay, let’s kick things off with the first part of our equation: 5⁷. Now, what does this mean? It simply means we're multiplying 5 by itself seven times. So, it’s 5 × 5 × 5 × 5 × 5 × 5 × 5. You might be tempted to reach for a calculator right away, and that's totally fine! But let's take a moment to appreciate the power of exponents. They’re a shorthand way of writing repeated multiplication, which comes in super handy in math and science.

When you punch 5⁷ into your calculator, you'll get a pretty big number: 78,125. That's right! Five to the power of seven is a whopping 78,125. We've conquered the first part! See, it's not so intimidating when we break it down, right? Remember, guys, in math, it's all about taking it one step at a time. Don’t rush, and make sure you understand each step before moving on. This is especially crucial when dealing with more complex problems. Now that we have 5⁷ sorted out, let's move on to the next part of our equation.

Part 2: Taming the Mixed Fraction – 3 ⁹⁄₁₂

Next up, we have 3 ⁹⁄₁₂. This is what we call a mixed fraction – a combination of a whole number (3) and a fraction (⁹⁄₁₂). To make things easier to work with, we need to convert this mixed fraction into an improper fraction. An improper fraction is simply a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number).

So, how do we do this conversion? It’s actually pretty straightforward. We multiply the whole number (3) by the denominator (12) and then add the numerator (9). This gives us our new numerator. The denominator stays the same. Let’s walk through it: (3 × 12) + 9 = 36 + 9 = 45. So, our new numerator is 45, and our denominator remains 12. This means 3 ⁹⁄₁₂ is equivalent to ⁴⁵⁄₁₂. But we're not quite done yet! We can simplify this fraction further.

Both 45 and 12 are divisible by 3. Dividing both the numerator and the denominator by 3 gives us ¹⁵⁄₄. This is our simplified improper fraction. Now, some of you might be thinking, “Why bother simplifying?” Well, simplifying fractions makes them easier to work with in calculations and also gives us a clearer picture of their value. So, remember, always look for opportunities to simplify! We've successfully converted and simplified our mixed fraction. Give yourselves a pat on the back! We’re making great progress.

Part 3: Unraveling the Radical – ⁹√49

Alright, guys, let's tackle the radical part of our equation: ⁹√49. Now, this might look a bit intimidating, but let's break it down. The symbol “√” is the radical symbol, and the small number 9 above it is the index of the radical. This means we're looking for the ninth root of 49. In simpler terms, we're trying to find a number that, when multiplied by itself nine times, equals 49.

But before we go hunting for the ninth root, let’s see if we can simplify 49 first. We know that 49 is 7 × 7, or 7². This is a crucial step! By recognizing that 49 is a perfect square (the square of an integer), we can rewrite our radical expression. So, ⁹√49 becomes ⁹√(7²). Now, this is where things get interesting. We can rewrite this radical expression using fractional exponents. Remember, the nth root of a number is the same as raising that number to the power of 1/n. So, ⁹√(7²) is the same as 7^(²/₉).

Unfortunately, 2/9 is not a nice round number, and 7^(²/₉) will result in a decimal approximation. So, for the sake of simplicity and to maintain accuracy, we will leave ⁹√49 as is for now and calculate its approximate value at the end. We've successfully unraveled the radical (at least for now!). We're getting closer to the finish line, guys!

Part 4: Putting It All Together – The Grand Finale

Okay, team, we've conquered the exponents, tamed the mixed fraction, and unraveled the radical. Now comes the exciting part – putting it all together! Our original equation was 5⁷ × 3 ⁹⁄₁₂ + ⁹√49. We've figured out that 5⁷ = 78,125, 3 ⁹⁄₁₂ = ¹⁵⁄₄, and ⁹√49 is approximately 1.2599 (using a calculator). So, let's substitute these values back into our equation: 78,125 × ¹⁵⁄₄ + ⁹√49.

Remember the order of operations, guys? We need to do multiplication before addition. So, let’s first calculate 78,125 × ¹⁵⁄₄. This is the same as 78,125 × 3.75, which equals 293,437.5. Now our equation looks like this: 293,437.5 + ⁹√49. We know ⁹√49 is approximately 1.2599, so we add that to 293,437.5. This gives us our final answer: 293,438.7599.

So, drumroll please… the solution to 5⁷ × 3 ⁹⁄₁₂ + ⁹√49 is approximately 293,438.7599! Woohoo! We did it! See, guys, even the most intimidating equations can be solved if you break them down into smaller, manageable steps. Remember to focus on understanding each step, and don't be afraid to ask for help when you need it. Math is a journey, not a race. And with a little practice and perseverance, you can conquer any mathematical mountain!

Final Thoughts: The Power of Breaking It Down

Congratulations, mathletes! You've successfully navigated a complex equation and emerged victorious. We started with what seemed like a daunting problem, 5⁷ × 3 ⁹⁄₁₂ + ⁹√49, but by breaking it down into smaller parts, we were able to tackle each component individually and arrive at the solution. This approach – breaking down a problem into smaller, more manageable steps – is a powerful strategy not just in math, but in life in general. Whether you're tackling a tough homework assignment, learning a new skill, or even planning a big project, remember the power of breaking it down.

Remember: In the realm of math, practice makes perfect. The more you practice, the more comfortable you'll become with different types of problems and the more confident you'll feel in your problem-solving abilities. So, keep exploring, keep questioning, and keep challenging yourselves. And most importantly, have fun with it! Math can be a fascinating and rewarding subject, and we hope this step-by-step guide has helped you unlock a little bit more of its magic. Keep up the awesome work, guys! You've got this!