Solving (7/8)⁴ A Step-by-Step Mathematical Guide
Navigating the world of mathematics often involves encountering expressions that seem perplexing at first glance. One such expression is (7/8)₄, which appears to combine fractions and exponents in a unique way. To truly understand what this means, we need to carefully dissect it and explore the underlying mathematical principles. This article serves as a comprehensive guide to demystifying this expression, delving into its meaning, and providing a step-by-step approach to solving it. Whether you are a student grappling with exponents and fractions, or simply a curious mind seeking to expand your mathematical horizons, this exploration will equip you with the knowledge and confidence to tackle similar challenges. We will break down the concept of exponents, particularly how they apply to fractions, and then apply this understanding to accurately calculate the value of (7/8)₄. Our journey will also touch upon the importance of order of operations and how it guides us in solving complex expressions. Ultimately, the goal is to not just provide the answer, but to foster a deeper comprehension of the mathematical concepts involved, empowering you to solve a wider range of problems independently.
The expression (7/8)₄ essentially asks us to raise the fraction 7/8 to the power of 4. In simpler terms, this means multiplying the fraction 7/8 by itself four times. It's crucial to recognize that the exponent applies to the entire fraction, both the numerator (7) and the denominator (8). A common mistake is to only apply the exponent to the numerator or the denominator, leading to an incorrect result. To avoid this pitfall, it is helpful to visualize the expression as (7/8) * (7/8) * (7/8) * (7/8). This clarifies that each component of the fraction is involved in the multiplication process. Understanding this fundamental concept is the key to correctly solving the problem. The next step involves applying the rules of fraction multiplication, which state that we multiply the numerators together and the denominators together. This process will lead us to a new fraction, which may or may not be in its simplest form. We will then explore how to simplify the resulting fraction, if necessary, to arrive at the final answer. By carefully following these steps, we can confidently solve (7/8)₄ and similar expressions involving fractions and exponents.
Understanding exponents is fundamental to solving (7/8)₄. An exponent, also known as a power, indicates how many times a base number is multiplied by itself. In the expression xⁿ, 'x' is the base, and 'n' is the exponent. For instance, 2³ means 2 multiplied by itself three times (2 * 2 * 2), which equals 8. When dealing with fractions raised to a power, the exponent applies to both the numerator and the denominator. This is a critical concept to grasp. Consider the fraction a/b raised to the power of n, written as (a/b)ⁿ. This is equivalent to (aⁿ) / (bⁿ). In other words, we raise both the numerator 'a' and the denominator 'b' to the power of 'n'. This principle is directly applicable to our problem, (7/8)₄. We will raise both 7 and 8 to the power of 4, which means multiplying 7 by itself four times (7 * 7 * 7 * 7) and multiplying 8 by itself four times (8 * 8 * 8 * 8). The results of these calculations will form the numerator and denominator of our final fraction. By understanding this rule of exponents applied to fractions, we can confidently move forward in solving the problem and arrive at the correct answer. This knowledge also extends to other similar problems, empowering you to tackle a wide range of mathematical expressions involving fractions and exponents.
Step-by-Step Solution for (7/8)₄
To successfully solve (7/8)₄, let's break down the process into a series of clear and manageable steps. This methodical approach will not only guide us to the correct answer but also enhance our understanding of the underlying mathematical principles involved. By following each step carefully, we minimize the chances of errors and gain a deeper appreciation for the logic behind the solution. This step-by-step methodology is invaluable for tackling more complex problems in the future. It provides a framework for analyzing the problem, identifying the necessary operations, and executing them in a logical sequence. In this section, we will meticulously work through each stage of the solution, ensuring clarity and accuracy at every step. Let's begin our journey to unravel the value of (7/8)₄.
Step 1: Applying the Exponent to the Numerator
Our first step in solving (7/8)₄ is to apply the exponent, which is 4, to the numerator, which is 7. This means we need to calculate 7 raised to the power of 4, or 7⁴. This is equivalent to multiplying 7 by itself four times: 7 * 7 * 7 * 7. To perform this calculation, we can break it down into smaller steps. First, we multiply 7 * 7, which gives us 49. Then, we multiply 49 by 7, which results in 343. Finally, we multiply 343 by 7 to get our final result for 7⁴. Accurate calculation at this stage is crucial, as it forms the numerator of our answer. A mistake here will propagate through the rest of the solution. Therefore, it is advisable to double-check the multiplication to ensure its accuracy. The result of this step, 7⁴, will be the numerator of our final fraction, representing the outcome of raising the numerator of the original fraction to the given power. This careful and methodical approach is essential for solving mathematical problems accurately.
Step 2: Applying the Exponent to the Denominator
Following our successful calculation of the numerator, our next step in solving (7/8)₄ involves applying the exponent, 4, to the denominator, which is 8. This means we need to calculate 8 raised to the power of 4, or 8⁴. Similar to the previous step, this is equivalent to multiplying 8 by itself four times: 8 * 8 * 8 * 8. We can break this calculation down into manageable parts. First, we multiply 8 * 8, which gives us 64. Then, we multiply 64 by 8, which results in 512. Finally, we multiply 512 by 8 to obtain our final result for 8⁴. Just as with the numerator calculation, accuracy is paramount here. The result of this step will form the denominator of our final fraction. A mistake in this calculation will lead to an incorrect final answer. Therefore, it is prudent to verify the multiplication to ensure its correctness. The value of 8⁴ will represent the denominator of our final fraction, showcasing the outcome of raising the denominator of the original fraction to the power of 4. This step, along with the previous step, demonstrates how exponents apply to both the numerator and the denominator of a fraction.
Step 3: Combining the Results
Having calculated both the numerator and the denominator raised to the power of 4, we now arrive at a crucial step in solving (7/8)₄: combining these results to form our final fraction. In the previous steps, we determined that 7⁴ equals 2401 and 8⁴ equals 4096. Now, we express these values as a fraction, with 7⁴ as the numerator and 8⁴ as the denominator. This gives us the fraction 2401/4096. This fraction represents the value of (7/8)₄ before any simplification. It is important to understand that this fraction is mathematically equivalent to the original expression. However, it may not be in its simplest form. The next step will involve examining this fraction to see if it can be reduced to its lowest terms. This involves finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by it. If the GCD is 1, then the fraction is already in its simplest form. This step of combining the results effectively brings together the calculations from the previous steps, providing us with a clear representation of the value of the original expression.
Step 4: Simplifying the Fraction (If Possible)
Our final step in solving (7/8)₄ involves simplifying the fraction we obtained in the previous step, 2401/4096. Simplifying a fraction means reducing it to its lowest terms while maintaining its value. This is achieved by dividing both the numerator and the denominator by their greatest common divisor (GCD). To determine if our fraction can be simplified, we need to find the GCD of 2401 and 4096. There are several methods to find the GCD, such as listing the factors of each number or using the Euclidean algorithm. In this case, the GCD of 2401 and 4096 is 1. This means that 2401 and 4096 have no common factors other than 1, and therefore, the fraction 2401/4096 is already in its simplest form. Consequently, no further simplification is possible. The fraction 2401/4096 represents our final answer to the problem (7/8)₄. This result demonstrates the importance of simplifying fractions whenever possible to express the answer in its most concise and understandable form. In this instance, the fraction was already in its simplest form, highlighting that not all fractions can be simplified.
Final Answer
After meticulously following each step, we have successfully unraveled the mystery of (7/8)₄. We started by understanding the meaning of exponents and how they apply to fractions. We then systematically calculated the numerator and denominator raised to the power of 4. Finally, we combined these results and attempted to simplify the fraction. Our journey has led us to the final answer, which is 2401/4096. This fraction represents the value of (7/8)₄ in its simplest form. It is the culmination of our step-by-step approach, demonstrating the power of breaking down complex problems into smaller, manageable steps. This solution not only provides the numerical answer but also reinforces our understanding of the mathematical principles involved. By carefully analyzing each step, we have gained a deeper appreciation for the relationship between exponents and fractions. This knowledge will serve as a valuable tool in tackling future mathematical challenges. The final answer, 2401/4096, stands as a testament to our methodical approach and our commitment to accuracy and clarity in mathematical problem-solving.
In conclusion, the expression (7/8)₄ is solved by raising both the numerator and the denominator to the power of 4. This results in the fraction 2401/4096, which is already in its simplest form. This comprehensive guide has walked you through each step of the solution, providing a clear and concise explanation of the underlying mathematical principles. We hope this has enhanced your understanding and equipped you with the confidence to tackle similar problems in the future.
