Evaluate 49^(3/2) Expression Without Exponents A Step-by-Step Guide

#Introduction

In mathematics, exponents play a crucial role in expressing repeated multiplication. However, exponents can also be fractions, leading to expressions that might seem complex at first glance. This article aims to provide a comprehensive guide on how to evaluate expressions with fractional exponents, focusing on the expression $49^{3/2}$ as an example. Understanding fractional exponents is essential for simplifying algebraic expressions, solving equations, and grasping more advanced mathematical concepts. We will break down the process step-by-step, ensuring clarity and ease of understanding.

Understanding Fractional Exponents

To effectively evaluate expressions with fractional exponents, it's essential to first understand what these exponents represent. A fractional exponent, such as $a^{m/n}$, can be interpreted in terms of roots and powers. The denominator n represents the index of a radical, while the numerator m represents the power to which the base is raised. Specifically, $a^{m/n}$ is equivalent to $\sqrt[n]{a^m}$ or $(\sqrt[n]{a})^m$. This means that we can either take the n-th root of a and then raise the result to the power of m, or we can raise a to the power of m and then take the n-th root. The order in which these operations are performed can sometimes simplify the calculation, especially when dealing with larger numbers. For instance, when dealing with perfect squares, cubes, or higher powers, it's often easier to take the root first to reduce the size of the numbers involved. Understanding this fundamental principle is the key to mastering the evaluation of expressions with fractional exponents. This concept is not just limited to numerical calculations; it extends to algebraic manipulations, where simplifying expressions often involves converting fractional exponents to radical forms or vice versa. The ability to fluently switch between these forms is a valuable skill in algebra and calculus. Moreover, fractional exponents are closely related to the concept of radicals, providing a bridge between exponential and radical notation. Grasping this connection enriches your understanding of mathematical operations and their interrelation. In the context of more advanced topics like calculus, fractional exponents appear frequently in differentiation and integration problems, making a solid foundation in this area crucial for success. Understanding fractional exponents also provides a deeper insight into the properties of exponents in general. For example, the rules of exponents, such as the product rule ($a^m \cdot a^n = a^{m+n}$) and the power rule ($(a^m)^n = a^{mn}$), also apply to fractional exponents. Knowing how to apply these rules correctly allows for further simplification of expressions and solving complex equations. In summary, fractional exponents are a fundamental concept in mathematics, bridging exponents and radicals and serving as a building block for more advanced topics. Mastering this concept equips you with essential skills for tackling a wide range of mathematical problems.

Evaluating $49^{3/2}$ Step-by-Step

Now, let's apply this understanding to evaluate the expression $49^{3/2}$. Following the principle that $a^{m/n} = (\sqrt[n]{a})^m$, we can rewrite $49^{3/2}$ as $(\sqrt{49})^3$. This transformation is crucial because it breaks down the problem into smaller, more manageable steps. First, we need to find the square root of 49. The square root of a number is a value that, when multiplied by itself, equals the original number. In this case, the square root of 49 is 7, since $7 \times 7 = 49$. So, $\sqrt{49} = 7$. Next, we need to raise this result to the power of 3. This means we need to calculate $7^3$, which is $7 \times 7 \times 7$. We already know that $7 \times 7 = 49$, so we just need to multiply 49 by 7. Multiplying 49 by 7 gives us 343. Therefore, $7^3 = 343$. Putting it all together, $49^{3/2} = (\sqrt{49})^3 = 7^3 = 343$. This step-by-step approach not only provides the solution but also illustrates the process of simplifying expressions with fractional exponents. By breaking down the problem, we avoid overwhelming calculations and reduce the chances of error. This method is applicable to a wide range of similar problems. Another way to approach this problem is to first raise 49 to the power of 3 and then take the square root. However, this method involves larger numbers and is generally more complex. For instance, $49^3$ is a significantly larger number than $7^3$, making the subsequent square root calculation more challenging. Therefore, it is often more efficient to take the root first and then raise to the power. This strategy is especially useful when dealing with larger bases and exponents. In addition to numerical calculations, understanding these steps is crucial for algebraic manipulations. When simplifying algebraic expressions with fractional exponents, a similar approach can be applied to break down complex expressions into simpler forms. For example, expressions involving variables raised to fractional powers can be simplified by converting them to radical forms or applying the properties of exponents. This step-by-step method ensures accuracy and provides a clear path to the solution, whether you are dealing with numbers or algebraic variables. By mastering this process, you build confidence and competence in handling fractional exponents.

Alternative Approach: $49^{3/2} = \sqrt{49^3}$

As mentioned earlier, there is an alternative approach to evaluate expressions with fractional exponents. Instead of calculating $(\sqrt{49})^3$, we can calculate $\sqrt{49^3}$. While this approach is mathematically equivalent, it often involves dealing with larger numbers, which can make the calculation more complex. Let's illustrate this method. First, we need to calculate $49^3$, which is $49 \times 49 \times 49$. We know that $49 \times 49 = 2401$, so we need to multiply 2401 by 49. This multiplication yields 117,649. Therefore, $49^3 = 117,649$. Next, we need to find the square root of 117,649. This is a significantly larger number than the 7 we obtained when taking the square root of 49 in the previous method. Finding the square root of 117,649 might require a calculator or a more involved manual calculation. However, if we perform the calculation, we find that $\sqrt{117,649} = 343$. Thus, using this alternative approach, we also arrive at the same answer: $49^{3/2} = \sqrt{49^3} = \sqrt{117,649} = 343$. This comparison highlights the efficiency of taking the root before raising to the power when dealing with fractional exponents. While both methods yield the correct answer, calculating the root first generally simplifies the process by reducing the size of the numbers involved. This is particularly beneficial when performing calculations manually or when dealing with numbers that do not have readily apparent square roots or higher roots. In the context of algebraic expressions, this principle remains valuable. Simplifying by taking the root first can often lead to more manageable expressions and easier algebraic manipulations. For example, when simplifying expressions involving variables raised to fractional powers, taking the root first can reduce the degree of the polynomial, making subsequent operations simpler. Therefore, while understanding both methods is important, prioritizing the root-first approach can save time and reduce the potential for errors. This approach aligns with best practices in mathematical simplification and contributes to a deeper understanding of the properties of exponents and radicals.

Final Answer: $49^{3/2} = 343$

In conclusion, we have successfully evaluated the expression $49^{3/2}$ using a step-by-step approach and explored an alternative method. By understanding the fundamental principles of fractional exponents, we can confidently tackle similar problems. The key takeaway is that a fractional exponent $a^{m/n}$ can be interpreted as $(\sqrt[n]{a})^m$ or $\sqrt[n]{a^m}$, and choosing the right order of operations can simplify the calculation. In the case of $49^{3/2}$, we found that $(\sqrt{49})^3 = 7^3 = 343$ and $\sqrt{49^3} = \sqrt{117,649} = 343$. Both methods lead to the same answer, but taking the root first proves to be the more efficient strategy in this case. Therefore, the final answer is $49^{3/2} = 343$. This process demonstrates the importance of understanding mathematical concepts and applying them strategically to solve problems efficiently. The ability to break down complex expressions into simpler components and to choose the most effective method is a hallmark of mathematical proficiency. Moreover, this exercise reinforces the connection between exponents and radicals, highlighting how they are two sides of the same mathematical coin. This understanding is crucial not only for simplifying numerical expressions but also for manipulating algebraic expressions and solving equations. As you continue your mathematical journey, mastering fractional exponents will serve as a valuable tool in your problem-solving arsenal. Remember to practice these techniques with a variety of examples to solidify your understanding and build confidence. With practice, you will find that evaluating expressions with fractional exponents becomes second nature, allowing you to tackle more advanced mathematical challenges with ease. So, keep exploring, keep practicing, and enjoy the power of mathematical thinking!

Therefore, the correct answer is:

A. $49^{3 / 2} = 343$