Simplifying Radical Expressions A Step By Step Guide

Hey there, math enthusiasts! Today, we're diving into the fascinating world of radical expressions. Radicals, those funky-looking symbols with roots and exponents, can seem daunting at first. But fear not! We're going to break down the process of simplifying them, so you can tackle even the trickiest problems with confidence. In this comprehensive guide, we'll dissect a specific problem involving radicals, explaining each step in detail and providing a solid foundation for your understanding. So, buckle up and let's get started on this mathematical adventure! Our main goal is to simplify the expression $2 \[3]{x^2} \cdot \sqrt{16 x}$, assuming $x > 0$. This problem beautifully combines cube roots and square roots, giving us an excellent opportunity to showcase various techniques for simplifying radical expressions. Before we jump into the solution, let's brush up on some essential properties of radicals and exponents. Remember, radicals are just another way of expressing exponents, particularly fractional exponents. Understanding this connection is crucial for manipulating and simplifying these expressions effectively. We'll be using properties like the product rule for radicals, which states that the nth root of a product is the product of the nth roots, and the power rule for exponents, which helps us combine exponents when raising a power to another power. These rules are the building blocks for our simplification journey. Let's begin by rewriting the radicals in terms of fractional exponents. This is often the first step towards simplifying complex radical expressions. By converting radicals to fractional exponents, we can leverage the familiar rules of exponents to combine and simplify terms. This approach provides a clear and systematic way to handle different roots and powers within the same expression.

Rewriting Radicals as Fractional Exponents

Okay, guys, let's kick things off by transforming our radicals into fractional exponents. This is a crucial step because it allows us to use the power of exponent rules to simplify the expression. So, let's break it down: The expression we're tackling is $2 \sqrt[3]{x^2} \cdot \sqrt{16 x}$. First, let's focus on the cube root, $\sqrt[3]{x^2}$. Remember that a cube root is the same as raising something to the power of $\frac{1}{3}$. So, $\sqrt[3]{x^2}$ can be rewritten as $(x^2)^{\frac{1}{3}}$. Now, let's handle the square root, $\sqrt{16 x}$. A square root is equivalent to raising something to the power of $\frac{1}{2}$. Thus, $\sqrt{16 x}$ can be rewritten as $(16 x)^{\frac{1}{2}}$. By making these conversions, we've transformed our radical expression into an exponential one, setting the stage for further simplification. This is a common technique in simplifying radical expressions because exponents are often easier to manipulate than radicals directly. The next step is to apply the power of a power rule, which states that $(a^m)^n = a^{mn}$. This rule allows us to simplify expressions where an exponent is raised to another exponent. In our case, we have $(x^2)^{\frac{1}{3}}$, and applying the power of a power rule will help us combine these exponents into a single, simplified exponent. This transformation is a key step in our journey to simplify the original expression. Once we've applied this rule, we'll have a clearer picture of the exponents involved and how they can be further combined. So, let's roll up our sleeves and apply the power of a power rule to keep moving forward! Remember, the goal here is to make the expression as simple and manageable as possible, and converting to fractional exponents is a major step in that direction. By the end of this section, you'll see how elegantly exponents can help us handle even the most intimidating radicals. Let's get to it!

H2: Applying Exponent Rules for Simplification

Alright, let's put those exponent rules to work and simplify things further! Remember, we've rewritten our expression as $2(x^2)^{\frac{1}{3}} \cdot (16 x)^{\frac{1}{2}}$. Our next move is to apply the power of a power rule to the term $(x^2)^{\frac{1}{3}}$. This rule tells us that when we raise a power to another power, we multiply the exponents. So, $(x^2)^{\frac{1}{3}}$ becomes $x^{2 \cdot \frac{1}{3}}$, which simplifies to $x^{\frac{2}{3}}$. See how we're making progress? Now, let's tackle the second part of our expression, $(16 x)^{\frac{1}{2}}$. Here, we need to remember another important exponent rule: the power of a product rule. This rule states that $(ab)^n = a^n b^n$. So, we can rewrite $(16 x)^{\frac{1}{2}}$ as $16^{\frac{1}{2}} \cdot x^{\frac{1}{2}}$. Now, we can simplify $16^{\frac{1}{2}}$. Remember that raising something to the power of $\frac{1}{2}$ is the same as taking its square root. The square root of 16 is 4, so $16^{\frac{1}{2}} = 4$. Therefore, $(16 x)^{\frac{1}{2}}$ simplifies to $4x^{\frac{1}{2}}$. Great job, guys! We've successfully applied exponent rules to break down our expression into simpler terms. Our expression now looks like $2 \cdot x^{\frac{2}{3}} \cdot 4 \cdot x^{\frac{1}{2}}$. We're getting closer to the finish line! The next step is to combine the constants and the terms with the same base (in this case, x). This will involve multiplying the constants together and using another exponent rule to combine the x terms. By carefully applying these rules, we're transforming a seemingly complex expression into something much more manageable. Remember, each step we take is a step closer to the simplified form. So, let's keep our focus, continue applying these powerful exponent rules, and watch as the expression reveals its true, simplified form.

H3: Combining Like Terms and Simplifying

Okay, let's bring it all together and simplify this expression once and for all! We've reached the point where our expression looks like this: $2 \cdot x^{\frac{2}{3}} \cdot 4 \cdot x^{\frac{1}{2}}$. The first thing we can do is combine the constants. We have 2 and 4, and multiplying them together gives us 8. So, our expression now becomes $8 \cdot x^{\frac{2}{3}} \cdot x^{\frac{1}{2}}$. Next, we need to combine the x terms. Remember the product of powers rule? It states that when you multiply terms with the same base, you add the exponents. In this case, we have $x^{\frac{2}{3}}$ and $x^{\frac{1}{2}}$, so we need to add the exponents $\frac{2}{3}$ and $\frac{1}{2}$. To add fractions, we need a common denominator. The least common multiple of 3 and 2 is 6, so we'll rewrite the fractions with a denominator of 6. $\frac{2}{3}$ becomes $\frac{4}{6}$, and $\frac{1}{2}$ becomes $\frac{3}{6}$. Now we can add them: $\frac{4}{6} + \frac{3}{6} = \frac{7}{6}$. So, $x^{\frac{2}{3}} \cdot x^{\frac{1}{2}}$ simplifies to $x^{\frac{7}{6}}$. Our expression now looks like $8x^{\frac{7}{6}}$. We're almost there! The final step is to convert this back into radical form. Remember that $x^{\frac{7}{6}}$ means the 6th root of x raised to the 7th power. We can rewrite this as $\sqrt[6]{x^7}$. But wait, we can simplify this radical further! Since 7 is greater than 6, we can pull out a whole x from the radical. We can rewrite $x^7$ as $x^6 \cdot x$. So, $\sqrt[6]{x^7}$ becomes $\sqrt[6]{x^6 \cdot x}$. The 6th root of $x^6$ is just x, so we can pull that out of the radical, leaving us with $x\sqrt[6]{x}$. Putting it all together, our simplified expression is $8x\sqrt[6]{x}$. Awesome job! We've successfully navigated through the world of radicals and exponents to simplify this expression. Remember, the key is to break down the problem into smaller, manageable steps, apply the rules of exponents and radicals, and keep simplifying until you reach the final answer.

H2: Conclusion: Mastering Radical Simplification

Woo-hoo! We did it! We successfully simplified the expression $2 \sqrt[3]{x^2} \cdot \sqrt{16 x}$ to $8x\sqrt[6]{x}$. That was quite the journey, but hopefully, you've gained a solid understanding of how to tackle similar problems. Simplifying radical expressions might seem challenging at first, but with practice and a firm grasp of the rules, you can conquer any radical that comes your way. Remember, the key is to break down the problem into smaller, manageable steps. Start by converting radicals to fractional exponents. This allows you to leverage the powerful rules of exponents, making the simplification process much smoother. Next, apply the exponent rules, such as the power of a power rule and the product of powers rule, to combine terms and simplify exponents. Don't forget to look for opportunities to simplify constants and pull out perfect roots from radicals. Finally, remember to convert your answer back into radical form if necessary. By following these steps and practicing regularly, you'll become a master of radical simplification in no time! And hey, if you ever get stuck, don't hesitate to review the steps we've covered in this guide or seek out additional resources. There's a whole world of math knowledge out there just waiting to be explored! Keep practicing, keep learning, and most importantly, keep having fun with math! You've got this!

The correct answer is A. $8 x \sqrt[6]{x}$.

H3: Step-by-step Solution

1. Convert radicals to fractional exponents:
   • $2 \sqrt[3]{x^2} \cdot \sqrt{16 x} = 2 x^{\frac{2}{3}} \cdot (16 x)^{\frac{1}{2}}$
2. Apply the power of a product rule:
   • $2 x^{\frac{2}{3}} \cdot (16 x)^{\frac{1}{2}} = 2 x^{\frac{2}{3}} \cdot 16^{\frac{1}{2}} \cdot x^{\frac{1}{2}}$
3. Simplify $16^{\frac{1}{2}}$:
   • $2 x^{\frac{2}{3}} \cdot 16^{\frac{1}{2}} \cdot x^{\frac{1}{2}} = 2 x^{\frac{2}{3}} \cdot 4 \cdot x^{\frac{1}{2}}$
4. Multiply the constants:
   • $2 x^{\frac{2}{3}} \cdot 4 \cdot x^{\frac{1}{2}} = 8 x^{\frac{2}{3}} \cdot x^{\frac{1}{2}}$
5. Apply the product of powers rule:
   • $8 x^{\frac{2}{3}} \cdot x^{\frac{1}{2}} = 8 x^{\frac{2}{3} + \frac{1}{2}}$
6. Add the exponents:
   • $8 x^{\frac{2}{3} + \frac{1}{2}} = 8 x^{\frac{7}{6}}$
7. Convert back to radical form:
   • $8 x^{\frac{7}{6}} = 8 x \sqrt[6]{x}$