Equivalent Expression Of Radical Form Of 13 Cubed Fifth Root
Introduction
In mathematics, understanding the relationship between radicals and fractional exponents is crucial for simplifying expressions and solving equations. This article delves into the equivalence of the expression $\sqrt[5]{13^3}$, exploring the fundamental principles that govern the conversion between radical and exponential forms. Our main focus will be to clearly explain how to transform a radical expression into its equivalent form using fractional exponents. This involves understanding the base, the exponent, and the index of the radical. We will dissect the given expression, identify its components, and then rewrite it in exponential form. This skill is not only essential for academic purposes but also has practical applications in various fields such as physics, engineering, and computer science, where mathematical expressions often need simplification for problem-solving.
Breaking Down Radicals and Exponents
When dealing with radicals and exponents, it's vital to grasp the core definitions. A radical expression involves a root symbol () and consists of two primary parts: the radicand (the number under the root) and the index (the degree of the root). For example, in the expression , 'a' is the radicand, and 'n' is the index. In the case of square roots, the index is implicitly 2 (i.e., is the same as ). Exponents, on the other hand, denote the number of times a base is multiplied by itself. An expression like $a^b$ indicates that 'a' is raised to the power of 'b', where 'a' is the base and 'b' is the exponent. Fractional exponents bridge the gap between radicals and exponents, offering a way to express roots as powers. A fractional exponent has the form $\frac{m}{n}$, where 'm' is the numerator (power) and 'n' is the denominator (root). The expression $a^{\frac{m}{n}}$ is equivalent to $\sqrt[n]{a^m}$. Understanding this equivalence is crucial for converting between radical and exponential forms, simplifying complex expressions, and solving equations involving roots and powers. Mastering these concepts lays a strong foundation for more advanced mathematical topics.
Converting Radicals to Fractional Exponents
Fractional exponents provide a powerful way to express radicals, streamlining mathematical operations and simplifying complex expressions. The key to converting a radical to a fractional exponent lies in recognizing the relationship between the index of the radical and the denominator of the fractional exponent. Let’s consider the general form of a radical expression: $\sqrt[n]a^m}$, where 'a' is the base, 'm' is the power, and 'n' is the index of the root. To convert this to a fractional exponent, we rewrite it as $a^{\frac{m}{n}}$. The index 'n' becomes the denominator of the fraction, while the power 'm' becomes the numerator. For instance, $\sqrt[3]{5^2}$ can be expressed as $5^{\frac{2}{3}}$. This conversion is not merely a notational change; it allows us to apply the rules of exponents to expressions involving radicals, making manipulations more straightforward. Consider another example$. Here, the base is 7, the power is implicitly 1 (since 7 is the same as $7^1$), and the index is 4. Therefore, this radical can be written as $7^{\frac{1}{4}}$. This method applies universally, allowing us to convert any radical expression into its equivalent fractional exponent form. Understanding and applying this conversion is a fundamental skill in algebra and calculus, facilitating the simplification of expressions and the solution of equations.
Analyzing the Given Expression: $\sqrt[5]{13^3}$
To effectively address the problem at hand, we must meticulously dissect the given expression, $\sqrt[5]{13^3}$, to identify its key components. This expression is in radical form, which means it consists of a radical symbol (), a radicand, and an index. The radicand is the value under the radical symbol, which in this case is $13^3$. This indicates that the base is 13, and it is raised to the power of 3. The index of the radical is the small number written above the radical symbol, signifying the degree of the root. Here, the index is 5, meaning we are taking the fifth root of $13^3$. Understanding these components is crucial for converting the radical expression into its equivalent form using fractional exponents. The base, 13, will remain the base in the exponential form. The power, 3, will become the numerator of the fractional exponent, and the index, 5, will become the denominator. This conversion process allows us to rewrite the expression in a more manageable form for mathematical operations. By carefully identifying and understanding each part of the radical expression, we set the stage for accurately transforming it into its equivalent exponential form, which is essential for solving mathematical problems involving radicals and exponents.
Applying the Conversion to $\sqrt[5]{13^3}$
Now that we have thoroughly analyzed the given expression, $\sqrt[5]{13^3}$, we can confidently apply the principles of converting radicals to fractional exponents. As we established, the expression consists of a base (13), a power (3), and an index (5). Following the rule that $\sqrt[n]{a^m}$ is equivalent to $a^{\frac{m}{n}}$, we can directly translate the radical form into its exponential counterpart. In this case, 'a' is 13, 'm' is 3, and 'n' is 5. Therefore, $\sqrt[5]{13^3}$ can be rewritten as $13^{\frac{3}{5}}$. This transformation is a straightforward application of the conversion rule, where the index of the radical becomes the denominator of the fractional exponent, and the power of the radicand becomes the numerator. This conversion not only simplifies the representation of the expression but also allows us to manipulate it using the rules of exponents, which can be particularly useful in more complex calculations. The fractional exponent form, $13^{\frac{3}{5}}$, is the equivalent representation of the original radical expression, making it easier to work with in various mathematical contexts.
Evaluating the Options
Having converted the given expression $\sqrt[5]{13^3}$ to its equivalent form $13^{\frac{3}{5}}$, we now need to evaluate the provided options to identify the correct answer. The options presented are:
A. 132 B. 1315 C. $13^{\frac{5}{3}}$ D. $13^{\frac{3}{5}}$
By comparing our converted expression, $13^{\frac{3}{5}}$, with the options, it becomes evident that option D, $13^{\frac{3}{5}}$, is the correct equivalent. Options A and B are numerical values and do not match the exponential form we derived. Option C, $13^{\frac{5}{3}}$, has the fractional exponent inverted, making it incorrect. The fractional exponent $\frac{3}{5}$ correctly represents the fifth root of $13^3$, with 3 as the power and 5 as the index of the root. Therefore, option D is the only option that accurately reflects the transformation of the radical expression into its fractional exponent form. This methodical evaluation underscores the importance of understanding the conversion process and applying it correctly to identify the equivalent expression from a set of choices. The ability to accurately convert and compare mathematical expressions is a fundamental skill in mathematics, enabling students and professionals to solve problems efficiently and effectively.
Conclusion
In conclusion, we have successfully transformed the radical expression $\sqrt[5]{13^3}$ into its equivalent fractional exponent form, $13^{\frac{3}{5}}$, through a systematic approach. This process involved understanding the relationship between radicals and exponents, identifying the components of the radical expression (base, power, and index), and applying the conversion rule $\sqrt[n]{a^m} = a^{\frac{m}{n}}$. By dissecting the expression and recognizing that the index of the root becomes the denominator of the fractional exponent while the power of the radicand becomes the numerator, we were able to accurately rewrite the expression in its exponential form. Furthermore, we meticulously evaluated the provided options and confirmed that option D, $13^{\frac{3}{5}}$, is the correct equivalent. This exercise highlights the importance of mastering the conversion between radicals and fractional exponents, a fundamental skill in mathematics that facilitates the simplification of expressions and the solution of equations. The ability to convert between radical and exponential forms enhances mathematical fluency and problem-solving capabilities, which are essential in various fields, including science, engineering, and technology. The understanding and application of these concepts not only improve mathematical proficiency but also foster a deeper appreciation for the interconnectedness of mathematical ideas.
