Simplify: $14(\sqrt[4]{a^5 B^2 C^4})-7 A C(\sqrt[4]{a B^2})$

by Andrew McMorgan 63 views

Hey math enthusiasts! Today, we're diving into simplifying a radical expression. This might look a bit intimidating at first, but don't worry, we'll break it down step by step. We're going to tackle the expression $14(\sqrt[4]{a^5 b^2 c^4})-7 a c(\sqrt[4]{a b^2})$, where $a \geq 0$ and $c \geq 0$. Let's get started!

Understanding the Problem

Before we jump into the solution, let's make sure we understand what the problem is asking. We have an expression involving radicals, specifically fourth roots, and our goal is to simplify it. This means we want to rewrite the expression in a cleaner, more concise form. We'll use properties of radicals and exponents to achieve this. Remember, the key here is to identify common factors and simplify the radicals as much as possible. Think of it like decluttering your mathematical space – we want everything neat and organized!

Breaking Down the Radicals

Okay, let's start by simplifying each radical term individually. The expression we're working with is $14(\sqrt[4]{a^5 b^2 c^4})-7 a c(\sqrt[4]{a b^2})$. We'll focus on the first term, $14(\sqrt[4]{a^5 b^2 c^4})$. The fourth root means we're looking for factors that appear four times inside the radical. We can rewrite $a^5$ as $a^4 imes a$ and $c^4$ is already a perfect fourth power. So, we have:

a5b2c44=a4imesaimesb2imesc44 \sqrt[4]{a^5 b^2 c^4} = \sqrt[4]{a^4 imes a imes b^2 imes c^4}

Now, we can take out the terms that have a fourth power:

a4imesaimesb2imesc44=acab24 \sqrt[4]{a^4 imes a imes b^2 imes c^4} = a c \sqrt[4]{a b^2}

So, the first term becomes:

14(a5b2c44)=14acab24 14(\sqrt[4]{a^5 b^2 c^4}) = 14 a c \sqrt[4]{a b^2}

See? We've already made progress! Now let's look at the second term.

Simplifying the Second Term

The second term in our expression is $7 a c(\sqrt[4]{a b^2})$. Guess what? This term is already in a pretty simplified form! The radical $(\sqrt[4]{a b^2})$ doesn't have any perfect fourth power factors we can extract. So, we can just leave it as is. This is actually good news because it means we're one step closer to combining the terms.

Combining Like Terms

Now that we've simplified both radical terms, let's bring them together. Our expression now looks like this:

14acab24−7acab24 14 a c \sqrt[4]{a b^2} - 7 a c \sqrt[4]{a b^2}

Notice anything familiar? We have two terms that both have the factor $a c \sqrt[4]{a b^2}$. This means they are like terms, and we can combine them just like we combine $2x$ and $3x$. We simply subtract the coefficients:

14acab24−7acab24=(14−7)acab24 14 a c \sqrt[4]{a b^2} - 7 a c \sqrt[4]{a b^2} = (14 - 7) a c \sqrt[4]{a b^2}

=7acab24 = 7 a c \sqrt[4]{a b^2}

And there you have it! We've simplified the expression.

The Final Simplified Form

After all that simplification, our final answer is:

7acab24 7 a c \sqrt[4]{a b^2}

This is the simplified form of the original expression $14(\sqrt[4]{a^5 b^2 c^4})-7 a c(\sqrt[4]{a b^2})$. It looks much cleaner and easier to work with, right? Remember, the key to simplifying radical expressions is to break them down into smaller parts, identify perfect powers, and combine like terms. You've nailed it!

Checking Our Work

It's always a good idea to double-check our work, especially in math. While we can't plug in every possible value for $a$, $b$, and $c$ (since they are variables), we can think about whether our answer makes sense. We started with an expression involving fourth roots, and our simplified answer still involves a fourth root, which is expected. We also combined the terms correctly, so the coefficients should be accurate. If you have a graphing calculator or software, you could even graph both the original expression and the simplified expression to see if they are the same. But for now, we can be confident that we've simplified the expression correctly. Great job, guys!

Key Takeaways

Before we wrap up, let's recap the key steps we took to simplify this expression. This will help you tackle similar problems in the future. Here are the main takeaways:

  1. Understand the Problem: Make sure you know what the question is asking. In this case, we needed to simplify a radical expression.
  2. Break Down the Radicals: Simplify each radical term individually by looking for perfect powers. In our example, we extracted $a^4$ and $c^4$ from the fourth root.
  3. Combine Like Terms: Identify terms with the same radical factor and combine their coefficients.
  4. Double-Check Your Work: Always review your steps and make sure your answer makes sense.

By following these steps, you can simplify even the most complex radical expressions. Keep practicing, and you'll become a pro in no time!

Practice Makes Perfect

Want to get even better at simplifying radical expressions? The best way is to practice! Try working through similar problems. You can find examples in textbooks, online resources, or even make up your own. Challenge yourself by varying the complexity of the expressions. For example, you could try simplifying expressions with different roots (like cube roots or fifth roots) or expressions with more variables. The more you practice, the more comfortable you'll become with the process. And remember, don't be afraid to make mistakes – they are part of the learning process. Just learn from them, and keep going!

Conclusion

So, there you have it! We've successfully simplified the expression $14(\sqrt[4]{a^5 b^2 c^4})-7 a c(\sqrt[4]{a b^2})$ to $7 a c \sqrt[4]{a b^2}$. We broke down the radicals, combined like terms, and even checked our work. You've learned valuable skills that you can use to simplify other radical expressions. Keep up the great work, and remember to have fun with math! You guys are awesome!