Solving $( \sqrt{ {7}^{3} } \times \sqrt[5]{ {7}^{2} } ) \div \sqrt[7]{ {7} } )$ A Step-by-Step Explanation

Hey guys! Math can sometimes feel like navigating a maze, right? Especially when you see expressions that look like $( \sqrt{ {7}^{3} } \times \sqrt[5]{ {7}^{2} } ) \div \sqrt[7]{ {7} } )$. But don't worry, we're going to break this down step by step and make it super clear. Think of this as a fun puzzle rather than a daunting task. We'll use some cool exponent rules and properties of radicals to simplify this expression. By the end of this guide, you'll not only know the answer but also understand the how and why behind it. Let's dive in and conquer this mathematical challenge together!

Understanding the Basics: Radicals and Exponents

Before we jump into the main problem, let's quickly refresh our understanding of radicals and exponents. Radicals, those expressions with the root symbol ($\sqrt{}$), are essentially another way of representing exponents. For example, $\sqrt{x}$ is the same as $x^{\frac{1}{2}}$. This connection is super important because it allows us to use the rules of exponents to simplify expressions involving radicals. Now, exponents themselves tell us how many times a number (the base) is multiplied by itself. So, $7^3$ means 7 multiplied by itself three times (7 * 7 * 7). The beauty of exponents lies in their consistent rules, which help us simplify complex expressions.

Converting Radicals to Exponents: The Key Step

The first step in simplifying our expression is to convert the radicals into exponents. This is crucial because it allows us to combine terms with the same base using exponent rules. Remember, the general rule is: $\sqrt[n]{a^m} = a^{\frac{m}{n}}$. This means the nth root of $a$ raised to the power of $m$ is the same as $a$ raised to the power of $\frac{m}{n}$. For instance, $\sqrt{7^3}$ can be rewritten as $7^{\frac{3}{2}}$. Similarly, $\sqrt[5]{7^2}$ becomes $7^{\frac{2}{5}}$, and $\sqrt[7]{7}$ is simply $7^{\frac{1}{7}}$. Once we've made these conversions, our original expression looks much friendlier: $(7^{\frac{3}{2}} \times 7^{\frac{2}{5}}) \div 7^{\frac{1}{7}}$. See? We've transformed radicals into exponents, and now we can leverage the power of exponent rules!

Multiplying with the Same Base: Adding Exponents

Now that we've got our radicals in exponent form, let's tackle the multiplication part: $7^{\frac{3}{2}} \times 7^{\frac{2}{5}}$. Here's where the exponent rules come to our rescue. When you're multiplying numbers with the same base, you simply add their exponents. Mathematically, this is expressed as: $a^m \times a^n = a^{m+n}$. So, in our case, we need to add the exponents $\frac{3}{2}$ and $\frac{2}{5}$. To add fractions, we need a common denominator. The least common multiple of 2 and 5 is 10, so we convert our fractions: $\frac{3}{2} = \frac{15}{10}$ and $\frac{2}{5} = \frac{4}{10}$. Now we can easily add them: $\frac{15}{10} + \frac{4}{10} = \frac{19}{10}$. This means $7^{\frac{3}{2}} \times 7^{\frac{2}{5}} = 7^{\frac{19}{10}}$. We've successfully simplified the multiplication part, and we're one step closer to the final answer!

Dividing with the Same Base: Subtracting Exponents

Next up, we need to deal with the division: $7^{\frac{19}{10}} \div 7^{\frac{1}{7}}$. Just like multiplication, division of numbers with the same base has a handy exponent rule: $a^m \div a^n = a^{m-n}$. This means we subtract the exponents. In our case, we need to subtract $\frac{1}{7}$ from $\frac{19}{10}$. Again, we need a common denominator. The least common multiple of 10 and 7 is 70. So, we convert our fractions: $\frac{19}{10} = \frac{133}{70}$ and $\frac{1}{7} = \frac{10}{70}$. Now we subtract: $\frac{133}{70} - \frac{10}{70} = \frac{123}{70}$. Therefore, $7^{\frac{19}{10}} \div 7^{\frac{1}{7}} = 7^{\frac{123}{70}}$. We're almost there! We've simplified the expression down to a single exponent.

The Grand Finale: Putting It All Together

Alright, let's recap! We started with the expression $( \sqrt{ {7}^{3} } \times \sqrt[5]{ {7}^{2} } ) \div \sqrt[7]{ {7} } )$. We converted radicals to exponents, used the multiplication rule to add exponents, and then used the division rule to subtract exponents. After all these steps, we've arrived at our simplified form: $7^{\frac{123}{70}}$. This is the answer! Now, you might be wondering if we can simplify this further. Well, $\frac{123}{70}$ is an improper fraction, meaning the numerator is larger than the denominator. We could convert it to a mixed number if we wanted to: $\frac{123}{70} = 1 \frac{53}{70}$. So, our answer could also be written as $7^{1 \frac{53}{70}}$, which is the same as $7^1 \times 7^{\frac{53}{70}}$ or $7 \times 7^{\frac{53}{70}}$. But, for the sake of simplicity, we'll stick with $7^{\frac{123}{70}}$ as our final, simplified answer. Great job, everyone! You've successfully navigated this mathematical puzzle!

Alternative Representation: Back to Radical Form

Just for kicks, let's see if we can express our answer, $7^{\frac{123}{70}}$, back in radical form. Remember, the exponent $\frac{m}{n}$ corresponds to the radical $\sqrt[n]{a^m}$. So, $7^{\frac{123}{70}}$ can be written as $\sqrt[70]{7^{123}}$. While this is mathematically equivalent to our previous answer, $7^{\frac{123}{70}}$, it's often preferred to leave the answer in exponent form when the exponent is a fraction. But it's good to know how to switch back and forth between these forms, just in case! It shows a deeper understanding of the relationship between exponents and radicals.

Key Takeaways: Mastering Exponents and Radicals

So, what have we learned today? The most important takeaway is the connection between radicals and exponents. By understanding that $\sqrt[n]{a^m} = a^{\frac{m}{n}}$, we can convert radical expressions into exponential ones and vice versa. This opens up a whole new world of simplification possibilities! We also refreshed our knowledge of exponent rules: when multiplying numbers with the same base, we add exponents; when dividing, we subtract exponents. These rules are fundamental in simplifying a wide range of mathematical expressions. Remember, the key to mastering these concepts is practice. The more problems you solve, the more comfortable you'll become with applying these rules and techniques. So, keep practicing, and you'll become a math whiz in no time!

Practice Makes Perfect: Try These Problems!

To really solidify your understanding, why not try a few practice problems? Here are a couple to get you started:

1. Simplify: $( \sqrt{ {5}^{5} } \times \sqrt[3]{ {5}^{2} } ) \div \sqrt[4]{ {5} } $
2. Simplify: $( \sqrt[3]{ {2}^{4} } \div \sqrt{ {2} } ) \times \sqrt[5]{ {2}^{3} } $

Try working through these problems using the same steps we used in this guide. Convert radicals to exponents, apply the multiplication and division rules, and simplify. Don't be afraid to make mistakes – that's how we learn! If you get stuck, revisit the concepts we discussed earlier. And remember, the more you practice, the easier it will become. Good luck, and happy simplifying!

Conclusion: You've Got This!

Wow, we've covered a lot in this guide! From understanding the basics of radicals and exponents to simplifying a complex expression, you've come a long way. Remember, the key to tackling math problems is to break them down into smaller, manageable steps. By converting radicals to exponents, applying exponent rules, and taking it one step at a time, even the most daunting expressions can be simplified. So, the next time you encounter a problem like $( \sqrt{ {7}^{3} } \times \sqrt[5]{ {7}^{2} } ) \div \sqrt[7]{ {7} } )$, don't panic! Remember the steps we've discussed, and you'll be well on your way to finding the solution. Keep practicing, keep learning, and most importantly, keep having fun with math! You've got this!