Simplifying 1250^(3/4) What Value Remains Under The Radical

Introduction

In the realm of mathematics, simplifying expressions to their most basic forms is a fundamental skill. This article delves into the process of simplifying the expression 1250^(3/4) into its simplest radical form. Our primary objective is to identify the value that remains under the radical sign after the simplification is complete. This exercise not only reinforces our understanding of exponents and radicals but also highlights the importance of prime factorization in simplifying mathematical expressions. We will embark on a step-by-step journey, breaking down the expression, applying relevant mathematical principles, and ultimately arriving at the solution. Along the way, we will explore the underlying concepts and techniques that make this simplification possible, ensuring a clear and comprehensive understanding for readers of all backgrounds.

Understanding the Fundamentals: Exponents and Radicals

Before we dive into the specifics of simplifying 1250^(3/4), let's establish a solid foundation by revisiting the fundamental concepts of exponents and radicals. Exponents represent the power to which a number is raised, indicating how many times the base number is multiplied by itself. For instance, in the expression 2^3, 2 is the base, and 3 is the exponent, signifying 2 multiplied by itself three times (2 * 2 * 2 = 8). Radicals, on the other hand, are the inverse operation of exponents. They represent the root of a number. The most common radical is the square root, denoted by the symbol √, which finds a number that, when multiplied by itself, equals the radicand (the number under the radical sign). For example, √9 = 3 because 3 * 3 = 9. However, radicals can also represent cube roots, fourth roots, and so on, denoted by an index number placed above the radical symbol (e.g., ∛8 represents the cube root of 8, which is 2). Understanding the relationship between exponents and radicals is crucial for simplifying expressions like 1250^(3/4), as it allows us to convert between these two forms and apply appropriate simplification techniques. The expression 1250^(3/4) can be interpreted as the fourth root of 1250 raised to the power of 3, or equivalently, the cube of the fourth root of 1250. This duality is key to our simplification strategy.

Breaking Down 1250: Prime Factorization

The first crucial step in simplifying 1250^(3/4) is to break down the base, 1250, into its prime factors. Prime factorization involves expressing a number as a product of its prime factors, which are prime numbers that divide the number exactly. This process helps us identify any perfect powers that can be extracted from under the radical sign. To find the prime factors of 1250, we can start by dividing it by the smallest prime number, 2: 1250 ÷ 2 = 625. Now we have 2 * 625. Next, we look at 625. It's not divisible by 2 or 3, but it is divisible by 5: 625 ÷ 5 = 125. So, we have 2 * 5 * 125. Continuing with 125, we divide by 5 again: 125 ÷ 5 = 25. This gives us 2 * 5 * 5 * 25. Finally, 25 is 5 * 5. Therefore, the prime factorization of 1250 is 2 * 5 * 5 * 5 * 5, which can be written as 2 * 5^4. This prime factorization is essential because it allows us to rewrite 1250^(3/4) as (2 * 5⁴⁾(3/4). This form is much easier to work with when simplifying the expression, as we can now apply the rules of exponents to each factor within the parentheses. By expressing 1250 in terms of its prime factors, we have laid the groundwork for extracting any perfect fourth powers, which will significantly simplify the radical expression.

Applying the Power of a Product Rule

With the prime factorization of 1250 in hand, we can now apply the power of a product rule to the expression (2 * 5⁴⁾(3/4). This rule states that (ab)^n = a^n * b^n, where a and b are any numbers, and n is an exponent. In our case, a = 2, b = 5^4, and n = 3/4. Applying this rule, we get (2 * 5⁴⁾(3/4) = 2^(3/4) * (5⁴⁾(3/4). This step is crucial because it separates the factors within the parentheses, allowing us to deal with each part individually. Now, we have two separate expressions to simplify: 2^(3/4) and (5⁴⁾(3/4). The first expression, 2^(3/4), represents 2 raised to the power of 3/4, which can be interpreted as the fourth root of 2 cubed. The second expression, (5⁴⁾(3/4), involves raising a power to another power. To simplify this, we use the power of a power rule, which states that (a^m)n = a^(m*n). Applying this rule, we get (5⁴⁾(3/4) = 5^(4 * 3/4) = 5^3. This simplification is significant because it removes the fractional exponent and leaves us with a whole number power of 5. The expression 5^3 is simply 5 * 5 * 5 = 125. Now, our original expression 1250^(3/4) has been transformed into 2^(3/4) * 125. This form is much simpler to understand and work with, as we have extracted the perfect fourth power of 5 from the radical. The next step is to express 2^(3/4) in its simplest radical form.

Simplifying the Radical: 2^(3/4)

Having simplified the expression to 2^(3/4) * 125, our next focus is on expressing 2^(3/4) in its simplest radical form. The fractional exponent 3/4 can be interpreted as a combination of a power and a root. The denominator, 4, indicates the root (in this case, the fourth root), and the numerator, 3, indicates the power to which the base is raised. Therefore, 2^(3/4) can be written as the fourth root of 2 cubed, which is √(4&2^3). To further simplify this, we first calculate 2^3, which is 2 * 2 * 2 = 8. So, we now have the fourth root of 8, written as √(4&8). This is the simplest radical form of 2^(3/4), as 8 has no perfect fourth power factors. In other words, there is no integer that, when raised to the power of 4, divides evenly into 8. Therefore, √(4&8) cannot be simplified further. Now, we can substitute this simplified radical form back into our expression. We had 1250^(3/4) = 2^(3/4) * 125. Replacing 2^(3/4) with its simplest radical form, √(4&8), we get 1250^(3/4) = 125 * √(4&8). This is the completely simplified form of the original expression. We have successfully extracted all possible perfect powers from the radical, leaving us with a whole number (125) multiplied by a simplified radical (√(4&8)). The final step is to identify the value that remains under the radical sign.

Identifying the Value Under the Radicand

After simplifying 1250^(3/4) to 125 * √(4&8), we can clearly see the value that remains under the radical sign. The radical part of the expression is √(4&8), which represents the fourth root of 8. The number under the radical sign, also known as the radicand, is 8. Therefore, the value that remains under the radical is 8. This concludes our simplification process. We started with the expression 1250^(3/4), broke it down using prime factorization and the rules of exponents and radicals, and ultimately arrived at the simplest radical form: 125 * √(4&8). The final step was to identify the radicand, which is the value under the radical sign. In this case, the radicand is 8. This exercise demonstrates the importance of understanding the relationship between exponents and radicals, as well as the power of prime factorization in simplifying mathematical expressions. By systematically breaking down the expression and applying the relevant mathematical principles, we were able to arrive at the solution in a clear and concise manner.

Conclusion

In summary, we have successfully simplified the expression 1250^(3/4) into its simplest radical form, which is 125 * √(4&8). Through a step-by-step process involving prime factorization, application of exponent rules, and simplification of radicals, we identified that the value remaining under the radical sign is 8. This exercise underscores the significance of mastering fundamental mathematical concepts such as exponents, radicals, and prime factorization. These skills are not only essential for simplifying expressions but also for solving a wide range of mathematical problems. The ability to break down complex expressions into simpler components and apply appropriate techniques is a hallmark of mathematical proficiency. By understanding the underlying principles and practicing regularly, anyone can develop the skills necessary to tackle similar challenges with confidence and accuracy. The journey of simplifying 1250^(3/4) serves as a valuable lesson in the power of mathematical reasoning and the importance of a solid foundation in fundamental concepts.

Therefore, the answer is 8.