Simplifying Radicals: Unveiling The Equivalent Expression

Hey math enthusiasts! Let's dive into a classic problem that tests your understanding of radicals and exponents. The question we're tackling today is: "Which of the following is equivalent to $\sqrt[5]{13^3}$?" This problem is a fantastic opportunity to brush up on some fundamental mathematical concepts, so let's get started. We'll explore the problem step-by-step to grasp the core concepts of simplifying radicals. Get ready to flex those math muscles and discover the correct answer! This problem beautifully showcases the relationship between radicals and exponents, and it's a piece of cake once you know the rules.

Understanding Radicals and Exponents

Alright, before we jump into the solution, let's make sure we're all on the same page. Remember that a radical, like the fifth root in our problem ($\sqrt[5]{}$), is just another way of expressing a fractional exponent. Think of it this way: when you see a radical, you can rewrite it as a number raised to a fractional power. Specifically, the nth root of a number can be expressed as that number raised to the power of 1/n. So, the fifth root of a number is the same as raising that number to the power of 1/5. Now, consider the expression inside the radical: $13^3$. This means 13 multiplied by itself three times. When we combine these two ideas, we're really close to solving our problem. The key is understanding how exponents work together. When you have an exponent raised to another exponent, you multiply the powers. For instance, $(x^a)^b = x^{a*b}$. This is a crucial rule for simplifying radicals. To make this super clear, let's break down how to convert our problem into an exponent form. Let's rewrite $\sqrt[5]{13^3}$. The fifth root is equivalent to the power of 1/5. The expression inside is $13^3$. So, the entire expression becomes $(13^3)^{1/5}$. Now, using the rule we mentioned above (multiply the exponents), we get $13^{3*(1/5)}$, which simplifies to $13^{3/5}$.

So, remember, guys, the main takeaway is to always keep in mind that radicals and exponents are besties. They are deeply interconnected, and knowing how to convert between them is the key to solving these types of problems. When you're faced with a radical, always consider converting it into an exponential form. This can make the problem way easier to solve. The next time you see a radical, try converting it to an exponential form, and you'll find it's a great strategy to simplify the problem and find the correct solution. Knowing this fundamental relationship will make you a math whiz in no time. This skill is invaluable for more advanced math topics. Keep practicing and you'll be acing these problems in no time! So, guys, take a moment to absorb this process, because we're about to put it to use and solve our original problem.

Step-by-Step Solution

Let's get down to the nitty-gritty and solve this problem together. We've got our expression: $\sqrt[5]{13^3}$. We already know that the fifth root can be written as a power of 1/5. So, let's rewrite the expression step by step. First, rewrite the radical as an exponent: $\sqrt[5]{13^3} = (13^3)^{1/5}$. Now, apply the power of a power rule: When you have an exponent raised to another exponent, you multiply the powers. In our case, this means multiplying 3 by 1/5. That gives us $3 * \frac{1}{5} = \frac{3}{5}$. Hence, $(13^3)^{1/5} = 13^{3/5}$. Thus, $\sqrt[5]{13^3} = 13^{3/5}$.

So, the answer is $13^{3/5}$.

Analyzing the Answer Choices

Let's take a look at the given answer choices and see which one matches our solution: A. $13^2$. This is incorrect because we simplified the original expression to $13^{3/5}$, not $13^2$. B. $13^{15}$. This is incorrect too. It arises from incorrectly multiplying the exponents. We need to multiply the exponent inside the radical (3) by the reciprocal of the root (1/5), not multiplying the root (5) by the exponent inside the radical (3). C. $13^{5/3}$. This is also incorrect. This option is derived from inverting the fraction we found in the exponent. D. $13^{3/5}$. This is the correct answer! We found that the equivalent expression of the original radical is $13$ raised to the power of $\frac{3}{5}$. So, congratulations, you've found the correct answer! Always go back and check your work to make sure you've got it right. It's easy to get mixed up with exponents, but with practice, it becomes second nature. Always remember the fundamental rules, especially the power of a power rule, and you'll always be in good shape.

Conclusion: Mastering Radicals and Exponents

We successfully tackled the problem of simplifying the radical $\sqrt[5]{13^3}$ and found its equivalent exponential form, which is $13^{3/5}$. This process involved understanding the relationship between radicals and exponents, and applying the power of a power rule. Remember, converting radicals to exponential form is a key strategy. This approach simplifies the expression and makes it easier to solve. Always double-check your work, particularly when dealing with fractional exponents, to avoid any mix-ups. This is a core concept, and being fluent in it will pave the way for tackling more complex mathematical challenges. Keep practicing these concepts, and you will become proficient in handling radicals and exponents. Understanding the rules and practicing them regularly will make you more confident. So, keep up the great work, and happy calculating!

In summary:

• $\sqrt[n]{x^m}$ is equivalent to $x^{\frac{m}{n}}$.
• When simplifying radicals, convert them into exponential form.
• Apply the power of a power rule to simplify the exponents.

By following these steps and remembering the key concepts, you can confidently solve similar problems. Keep practicing and exploring, and your math skills will continue to grow! And always, always double-check your work to catch those tricky little mistakes. Remember, practice makes perfect. Keep up the great work, guys!