Understanding And Simplifying The Fifth Root Of 13 Cubed

Hey guys! Let's dive into a super interesting math problem today that involves understanding exponents and radicals. We're going to break down the expression ${\sqrt[5]{13^3}}$ and figure out which of the given options is equivalent to it. This is a classic example of how exponents and radicals are just different ways of expressing the same mathematical idea. So, grab your thinking caps, and let's get started!

Understanding the Basics: Exponents and Radicals

To kick things off, let's quickly recap what exponents and radicals are all about. Exponents are a shorthand way of showing repeated multiplication. For instance, ${13^3}$ means 13 multiplied by itself three times: ${13 \times 13 \times 13}$. The '3' here is the exponent, and it tells us how many times to multiply the base (which is 13) by itself.

On the flip side, radicals are used to find the root of a number. The most common radical is the square root, denoted by ${\sqrt{\ }}$, which asks: "What number, when multiplied by itself, gives you the number under the root?" For example, ${\sqrt{25} = 5}$ because ${5 \times 5 = 25}$.

But it doesn't stop at square roots! We can also have cube roots, fourth roots, fifth roots, and so on. The number in the crook of the radical symbol (like the '5' in ${\sqrt[5]{\ }}$) is called the index, and it tells us what kind of root we're looking for. So, ${\sqrt[5]{32}}$ asks: "What number, when multiplied by itself five times, gives you 32?" The answer here is 2, because ${2 \times 2 \times 2 \times 2 \times 2 = 32}$.

The Connection: Radicals as Fractional Exponents

Now, here's where things get really cool: radicals can actually be expressed as fractional exponents. This is a crucial concept for simplifying expressions like the one we're tackling today. The general rule is:

${\sqrt[n]{a^m} = a^{\frac{m}{n}}}$

In plain English, this means that the ${n}$-th root of ${a}$ raised to the power of ${m}$ is the same as ${a}$ raised to the power of ${\frac{m}{n}}$. The index of the radical (${n}$) becomes the denominator of the fraction, and the exponent inside the radical (${m}$) becomes the numerator.

This connection between radicals and fractional exponents is super powerful because it allows us to use the rules of exponents to simplify expressions involving radicals, and vice versa. For instance, if we have ${\sqrt[3]{x^2}}$, we can rewrite it as ${x^{\frac{2}{3}}}$. Similarly, if we see ${y^{\frac{1}{2}}}$, we know it's the same as ${\sqrt{y}}$ (the square root of ${y}$, since the index is 2 when it's not explicitly written).

Applying the Concept to Our Problem: $\sqrt[5]{13^3}$

Okay, with the basics covered, let's get back to our original problem: finding an equivalent expression for ${\sqrt[5]{13^3}}$. We're going to use the rule we just discussed about radicals and fractional exponents to rewrite this expression.

Looking at ${\sqrt[5]{13^3}}$, we can identify the different parts:

• The base is 13.
• The exponent inside the radical is 3.
• The index of the radical (the little number outside the radical symbol) is 5.

Using our rule {\sqrt[n]{a^m} = a^{\frac{m}{n}}\, we can rewrite \(\sqrt[5]{13^3}} as ${13}$ raised to a fractional power. The exponent inside the radical (3) becomes the numerator of the fraction, and the index of the radical (5) becomes the denominator. So, we get:

${\sqrt[5]{13^3} = 13^{\frac{3}{5}}}$

And that's it! We've successfully converted the radical expression into an equivalent expression with a fractional exponent. This is a prime example of how understanding the relationship between radicals and exponents can help simplify mathematical expressions.

Evaluating the Options

Now that we know ${\sqrt[5]{13^3}}$ is the same as (13^{\frac{3}{5}}, let's look at the options provided and see which one matches:

1. ${13^2}$: This is simply 13 squared, which is ${13 \times 13 = 169}$. This is not the same as ${13^{\frac{3}{5}}}$.
2. ${13^{15}}$: This is 13 raised to the power of 15, which is a huge number. Definitely not equivalent to ${13^{\frac{3}{5}}}$.
3. ${13^{\frac{5}{3}}}$: This is 13 raised to the power of ${\frac{5}{3}}$. Notice that the fraction is flipped compared to what we found. So, this is not the correct answer.
4. ${13^{\frac{3}{5}}}$: Ah, here we go! This is exactly what we derived using the rule for converting radicals to fractional exponents. This is the correct equivalent expression.

So, the correct answer is ${13^{\frac{3}{5}}}$. We found this by understanding the fundamental relationship between radicals and exponents, and by carefully applying the rule for converting between the two forms. This kind of problem really highlights the importance of mastering these basic mathematical concepts!

Diving Deeper: Why This Matters

Okay, so we've cracked the problem, but you might be wondering, "Why does all this matter?" Well, understanding how to manipulate exponents and radicals is crucial in many areas of mathematics and beyond. These concepts pop up in algebra, calculus, physics, engineering, and even computer science.

For instance, in physics, you might encounter radicals when dealing with formulas involving square roots, like calculating the speed of an object or the period of a pendulum. In engineering, exponents and radicals are used in calculations related to electrical circuits, structural mechanics, and fluid dynamics. And in computer science, these concepts are fundamental to understanding algorithms, data structures, and cryptography.

Real-World Applications of Exponents and Radicals

Let's look at a few specific examples of how exponents and radicals are used in the real world:

• Finance: Compound interest calculations involve exponents. The formula for compound interest is ${A = P(1 + \frac{r}{n})^{nt}}$, where ${A}$ is the final amount, ${P}$ is the principal, ${r}$ is the interest rate, ${n}$ is the number of times interest is compounded per year, and ${t}$ is the number of years. The exponent ${nt}$ plays a key role in determining how quickly your investment grows.
• Population Growth: Exponential functions are used to model population growth. If a population grows at a constant rate, its size over time can be modeled by an exponential function. This helps demographers and policymakers make predictions about future population trends.
• Radioactive Decay: The decay of radioactive isotopes follows an exponential pattern. The half-life of an isotope (the time it takes for half of the substance to decay) is related to an exponential decay constant. This is used in carbon dating and nuclear medicine.
• Sound and Light: The intensity of sound and light decreases with distance according to an inverse square law, which involves exponents. This is why a sound seems quieter as you move farther away from the source, and why a light appears dimmer.
• Computer Graphics: Exponents and radicals are used in computer graphics to perform transformations like scaling, rotation, and translation. They are also used in lighting calculations and texture mapping.

Tips for Mastering Exponents and Radicals

So, how can you become a pro at working with exponents and radicals? Here are a few tips:

1. Practice, practice, practice: The more you work with exponents and radicals, the more comfortable you'll become. Try solving a variety of problems, from simple simplifications to more complex equations.
2. Memorize the rules: Knowing the rules of exponents and the relationship between radicals and fractional exponents is essential. Make flashcards or create a cheat sheet to help you remember them.
3. Understand the concepts: Don't just memorize the rules; make sure you understand why they work. This will help you apply them in different situations and avoid common mistakes.
4. Break down complex problems: If you're faced with a challenging problem, try breaking it down into smaller, more manageable steps. This can make the problem seem less daunting and help you identify the best approach.
5. Use online resources: There are tons of great resources online, including tutorials, videos, and practice problems. Take advantage of these resources to supplement your learning.

Wrapping Up

Alright, guys, that's a wrap! We've explored the fascinating world of exponents and radicals, tackled a tricky problem involving the fifth root of 13 cubed, and discussed why these concepts are so important. Remember, the key to mastering math is understanding the fundamental principles and practicing regularly. So, keep exploring, keep questioning, and keep learning!

If you have any more questions or want to dive deeper into this topic, feel free to ask. Happy math-ing!