Simplifying $2x^5 \sqrt{x} + 3\sqrt{x^{11}}$: A Step-by-Step Guide

Hey guys! Today, we're going to dive into simplifying the expression $2x^5 \sqrt{x} + 3\sqrt{x^{11}}$. This might look a bit intimidating at first glance, but don't worry, we'll break it down step by step so it's super easy to understand. We'll cover the basics of radical expressions, exponent rules, and how to combine like terms. By the end of this guide, you'll be a pro at simplifying expressions like this! So, let's get started and make math a little less scary and a lot more fun!

Understanding the Basics

Before we jump into the problem, let's quickly refresh some key concepts. We're dealing with radical expressions and exponents, so having a solid grasp on these will make the process much smoother. Think of this as our pre-game warm-up before the main event!

Radical Expressions

First off, what's a radical expression? Simply put, it's an expression that includes a radical symbol, which looks like this: $\sqrt{}$. The most common type is the square root, where we're looking for a number that, when multiplied by itself, equals the number under the radical. For example, $\sqrt{9} = 3$ because 3 * 3 = 9. The number inside the radical is called the radicand. We also have cube roots ($\sqrt[3]{}$), fourth roots ($\sqrt[4]{}$), and so on. Understanding how radicals work is crucial for simplifying expressions, so make sure you're comfortable with the basics.

Exponent Rules

Next up, exponents! Exponents tell us how many times to multiply a number by itself. For example, $x^3$ means x * x * x. There are a few key exponent rules we'll use:

• Product of Powers: $x^a * x^b = x^{a+b}$. When multiplying terms with the same base, you add the exponents.
• Power of a Power: $(x^a)^b = x^{a*b}$. When raising a power to another power, you multiply the exponents.
• Fractional Exponents: $x^{\frac{a}{b}} = \sqrt[b]{x^a}$. A fractional exponent represents both a power and a root. The numerator is the power, and the denominator is the index of the root. This is super important for dealing with radicals!

Knowing these rules is like having the secret code to unlock the problem. Keep them handy as we move forward.

Breaking Down the Expression

Okay, now that we've warmed up with the basics, let's tackle the expression $2x^5 \sqrt{x} + 3\sqrt{x^{11}}$. The first step is to rewrite the radical terms using fractional exponents. This will make it much easier to apply our exponent rules and simplify the expression.

Rewriting Radicals as Fractional Exponents

Remember that fractional exponent rule we just talked about? $x^{\frac{a}{b}} = \sqrt[b]{x^a}$. Let's use it to rewrite our radical terms.

• For the first term, we have $\sqrt{x}$. This is the same as $x^{\frac{1}{2}}$ because the square root is the same as raising to the power of $\frac{1}{2}$.
• For the second term, we have $\sqrt{x^{11}}$. This can be rewritten as $x^{\frac{11}{2}}$.

Now our expression looks like this: $2x^5 x^{\frac{1}{2}} + 3x^{\frac{11}{2}}$. See how much cleaner that looks already? We've transformed the radicals into exponents, which opens up a whole new world of simplification possibilities.

Simplifying the Terms

Now that we've rewritten our expression with fractional exponents, we can start simplifying each term individually. We'll use the exponent rules we discussed earlier to combine the exponents and make the expression more manageable. This is where the real magic happens, guys!

Simplifying the First Term: $2x^5 x^{\frac{1}{2}}$

Let's focus on the first term: $2x^5 x^{\frac{1}{2}}$. We have the same base (x) multiplied together, so we can use the Product of Powers rule: $x^a * x^b = x^{a+b}$.

In this case, we have $x^5 * x^{\frac{1}{2}}$. We need to add the exponents: 5 + $\frac{1}{2}$. To do this, we need a common denominator. We can rewrite 5 as $\frac{10}{2}$, so we have $\frac{10}{2} + \frac{1}{2} = \frac{11}{2}$.

Therefore, $x^5 * x^{\frac{1}{2}} = x^{\frac{11}{2}}$. Now, let's put it back into our term: $2x^5 x^{\frac{1}{2}} = 2x^{\frac{11}{2}}$. Great job! We've simplified the first term.

Simplifying the Second Term: $3x^{\frac{11}{2}}$

Now let's look at the second term: $3x^{\frac{11}{2}}$. Guess what? It's already in its simplest form! There's nothing more we can do to this term on its own. Sometimes, things are easier than they seem, right?

Combining Like Terms

We've simplified each term individually, and now it's time to combine them. But before we can do that, we need to make sure we're dealing with like terms. Like terms are terms that have the same variable raised to the same power. This is like making sure we're comparing apples to apples and not apples to oranges.

Identifying Like Terms

Our simplified expression is $2x^{\frac{11}{2}} + 3x^{\frac{11}{2}}$. Notice anything similar? Both terms have the same variable, x, raised to the same power, $\frac{11}{2}$. That means they are indeed like terms! This is fantastic news because it means we can combine them easily.

Combining the Terms

To combine like terms, we simply add their coefficients. The coefficient is the number in front of the variable. In our expression, the coefficients are 2 and 3. So, we add them together: 2 + 3 = 5.

Now, we just write the new coefficient in front of the common term: $5x^{\frac{11}{2}}$. And just like that, we've combined the like terms!

Converting Back to Radical Form (Optional)

We've successfully simplified the expression using fractional exponents, but sometimes it's helpful to convert back to radical form, especially if the original problem was given in radical form. It's like speaking the same language as the original problem.

Rewriting the Exponent as a Radical

Remember our fractional exponent rule? $x^{\frac{a}{b}} = \sqrt[b]{x^a}$. We'll use this rule in reverse to rewrite $x^{\frac{11}{2}}$ as a radical.

The denominator of the fraction, 2, becomes the index of the radical (which means it's a square root). The numerator, 11, becomes the power of x inside the radical. So, $x^{\frac{11}{2}} = \sqrt{x^{11}}$.

Now our expression looks like this: $5\sqrt{x^{11}}$.

Further Simplification of the Radical

We're not quite done yet! We can simplify $\sqrt{x^{11}}$ a bit further. We need to find the largest even power of x that is less than or equal to 11. In this case, it's $x^{10}$. We can rewrite $x^{11}$ as $x^{10} * x$.

So, $\sqrt{x^{11}} = \sqrt{x^{10} * x}$. Now, we can use the property $\sqrt{ab} = \sqrt{a} * \sqrt{b}$ to separate the radical: $\sqrt{x^{10} * x} = \sqrt{x^{10}} * \sqrt{x}$.

The square root of $x^{10}$ is $x^5$ (because $\sqrt{x^{10}} = x^{\frac{10}{2}} = x^5$). So, $\sqrt{x^{10}} * \sqrt{x} = x^5\sqrt{x}$.

Finally, we substitute this back into our expression: $5\sqrt{x^{11}} = 5x^5\sqrt{x}$.

The Final Simplified Expression

We've done it! After all those steps, the fully simplified expression is: $5x^5\sqrt{x}$.

Let's recap the entire process:

1. Rewrote radicals as fractional exponents.
2. Simplified each term using exponent rules.
3. Combined like terms.
4. Converted back to radical form (optional).
5. Further simplified the radical.

Conclusion

Simplifying expressions like $2x^5 \sqrt{x} + 3\sqrt{x^{11}}$ might seem daunting initially, but by breaking it down into manageable steps and understanding the underlying principles, it becomes much less intimidating. Remember, guys, the key is to take it one step at a time, apply the rules you know, and don't be afraid to practice. The more you practice, the more confident you'll become!

I hope this guide has helped you understand how to simplify radical expressions. Keep practicing, and you'll be a math whiz in no time! If you have any questions or want to try another example, feel free to ask. Happy simplifying!