Simplifying (15/16)^4 × (8/15)^4 × (16/121)^4 A Detailed Solution

This article delves into the simplification and solution of the mathematical expression (15/16)^4 × (8/15)^4 × (16/121)^4. We will explore the fundamental concepts of exponents and fractions, applying them step-by-step to arrive at the final answer. This exercise not only strengthens our understanding of mathematical operations but also highlights the beauty of simplification in complex expressions. Let's embark on this mathematical journey together!

Breaking Down the Expression: A Step-by-Step Approach

In this section, we will dissect the given expression, focusing on simplifying each component before performing the multiplication. Our primary tool here is the understanding of exponents and how they interact with fractions. An exponent indicates how many times a number (the base) is multiplied by itself. When dealing with fractions raised to a power, the exponent applies to both the numerator and the denominator. This property allows us to rewrite and simplify the expression effectively.

Let's start by examining the first term, (15/16)^4. This signifies that the fraction 15/16 is multiplied by itself four times. Applying the exponent rule for fractions, we can express this as (15^4) / (16^4). While we could calculate these large powers directly, it's often more strategic to look for opportunities to simplify before diving into computations. We'll keep this in mind as we move forward.

Next, we turn our attention to the second term, (8/15)^4. Similar to the first term, this can be rewritten as (8^4) / (15^4). Notice that we now have 15^4 in both the numerator of the first term and the denominator of the second term. This is a crucial observation, as it suggests a potential for simplification through cancellation. Cancellation is a fundamental principle in fraction manipulation, allowing us to reduce fractions to their simplest form by dividing both the numerator and denominator by a common factor.

Finally, we consider the third term, (16/121)^4. This term can be expressed as (16^4) / (121^4). Again, we observe a potential for simplification. We have 16^4 in the denominator of the first term and the numerator of this third term. This further reinforces the idea that we can significantly simplify the expression before performing any heavy calculations.

By breaking down the expression into its individual components and applying the basic principles of exponents and fractions, we've set the stage for a more streamlined simplification process. In the following sections, we will leverage these observations to cancel out common factors and arrive at the solution.

Leveraging Exponent Rules and Fraction Simplification

The heart of simplifying this mathematical expression lies in the strategic application of exponent rules and fraction simplification techniques. As we've already established, an exponent indicates repeated multiplication, and when a fraction is raised to a power, both the numerator and the denominator are raised to that power. Fraction simplification involves identifying common factors in the numerator and denominator and canceling them out. By combining these two powerful tools, we can efficiently solve the given problem.

Recall that our expression is (15/16)^4 × (8/15)^4 × (16/121)^4. We've already rewritten this as (15^4 / 16^4) × (8^4 / 15^4) × (16^4 / 121^4). Now, the beauty of this form becomes apparent. We have 15^4 in both the numerator of the first fraction and the denominator of the second fraction. This allows us to cancel out 15^4 completely, leaving us with a simplified expression.

Similarly, we observe that 16^4 appears in the denominator of the first fraction and the numerator of the third fraction. This means we can also cancel out 16^4. This cancellation significantly reduces the complexity of the expression, demonstrating the power of identifying and eliminating common factors. After these cancellations, our expression transforms into (1) × (8^4) × (1 / 121^4), which is simply 8^4 / 121^4.

Now, let's focus on simplifying 8^4 and 121^4. We know that 8 is 2 cubed (2^3), so 8^4 can be written as (2³⁾4. Using the power of a power rule, which states that (a^m)n = a^(mn), we can simplify (2³⁾4 to 2^(34), which is 2^12. This is a much more manageable number to work with.

Next, we consider 121^4. We recognize that 121 is 11 squared (11^2), so 121^4 can be expressed as (11²⁾4. Applying the power of a power rule again, we get 11^(2*4), which simplifies to 11^8. Now our expression is 2^12 / 11^8. This transformation highlights the elegance of using prime factorization and exponent rules to reduce complex numbers into simpler forms.

In the next section, we will calculate these final powers and express the result as a simplified fraction. The strategic use of exponent rules and fraction simplification has brought us closer to the solution, demonstrating the importance of these techniques in mathematical problem-solving.

Calculating the Final Result: Expressing the Solution

Having simplified the expression to 2^12 / 11^8, we are now in the final stage of calculating the numerical result. This involves computing the values of 2 raised to the power of 12 and 11 raised to the power of 8. While these might seem like large numbers, they are straightforward to calculate, especially with the aid of a calculator or by breaking them down into smaller multiplications.

Let's start with 2^12. This means 2 multiplied by itself 12 times. We can compute this as 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2. Alternatively, we can break it down into smaller powers. For example, we know that 2^10 is 1024. Therefore, 2^12 is 2^10 * 2^2, which is 1024 * 4. Multiplying 1024 by 4 gives us 4096. So, 2^12 equals 4096.

Now, let's calculate 11^8. This means 11 multiplied by itself 8 times. This calculation is a bit more involved, but we can still approach it systematically. We know that 11^2 is 121. Therefore, 11^4 is 121 * 121, which equals 14641. Then, 11^8 is 14641 * 14641. Performing this multiplication yields 214358881. This demonstrates how quickly powers can grow, even with relatively small base numbers.

Therefore, our expression 2^12 / 11^8 becomes 4096 / 214358881. This is the simplified fractional representation of the original expression. We have successfully navigated through the simplification process, leveraging exponent rules, fraction simplification, and strategic calculations to arrive at this final answer. The result is a fraction where the numerator is 4096 and the denominator is 214358881. This fraction is in its simplest form, as 4096 and 214358881 share no common factors other than 1.

In conclusion, by meticulously applying mathematical principles and techniques, we have transformed a seemingly complex expression into a manageable and easily understandable solution. This process highlights the importance of breaking down problems into smaller steps, identifying opportunities for simplification, and leveraging fundamental mathematical rules to achieve accurate results.

Alternative Approaches and Insights

While we've presented a step-by-step solution to the expression (15/16)^4 × (8/15)^4 × (16/121)^4, it's valuable to consider alternative approaches and gain further insights into the problem. This not only reinforces our understanding but also enhances our problem-solving skills. Different perspectives can often reveal hidden patterns or lead to more efficient solutions.

One alternative approach involves recognizing the structure of the expression from the outset. We can group the terms with the same exponent, allowing us to rewrite the expression as [(15/16) × (8/15) × (16/121)]^4. This initial regrouping highlights the possibility of simplifying the fractions within the brackets before raising the entire product to the fourth power. This approach can be particularly beneficial for those who prefer to minimize the calculations involving large numbers.

Inside the brackets, we have the product of three fractions: (15/16) × (8/15) × (16/121). We can simplify this product by canceling common factors across the numerators and denominators. Notice that the 15 in the numerator of the first fraction cancels with the 15 in the denominator of the second fraction. Similarly, the 16 in the denominator of the first fraction cancels with the 16 in the numerator of the third fraction. This leaves us with 8/121 inside the brackets. Therefore, the expression becomes (8/121)^4.

Now, we need to raise 8/121 to the fourth power. This means we have (8^4) / (121^4). As we calculated earlier, 8^4 is (2³⁾4, which is 2^12, and that equals 4096. Also, 121^4 is (11²⁾4, which is 11^8, and that equals 214358881. So, we arrive at the same simplified fraction, 4096 / 214358881, but through a slightly different route. This demonstrates that there can be multiple pathways to the same correct answer in mathematics.

Another valuable insight is to recognize the prime factorization of the numbers involved early in the process. Breaking down 15, 16, 8, and 121 into their prime factors can provide a clearer picture of the potential for simplification. For instance, 15 is 3 × 5, 16 is 2^4, 8 is 2^3, and 121 is 11^2. By substituting these prime factorizations into the original expression, we can directly apply exponent rules and identify common factors for cancellation. This approach is particularly helpful for complex expressions with multiple terms and large numbers.

By exploring alternative approaches and insights, we not only reinforce our understanding of the specific problem but also develop a more flexible and adaptable problem-solving mindset. The ability to view a problem from different angles and identify the most efficient solution path is a hallmark of mathematical proficiency.

Conclusion: The Power of Mathematical Simplification

In conclusion, the journey of simplifying the mathematical expression (15/16)^4 × (8/15)^4 × (16/121)^4 has been a testament to the power and elegance of mathematical principles. We've demonstrated how the strategic application of exponent rules, fraction simplification techniques, and prime factorization can transform a seemingly complex expression into a manageable and easily understandable solution. This exercise underscores the importance of breaking down problems into smaller, more manageable steps, identifying opportunities for simplification, and leveraging fundamental mathematical rules to achieve accurate results.

Throughout this article, we've emphasized the value of a step-by-step approach. By first dissecting the expression into its individual components, we were able to identify opportunities for simplification. We then skillfully applied exponent rules, such as the power of a power rule, and fraction simplification techniques, such as canceling common factors, to reduce the complexity of the expression. This methodical approach not only made the problem more approachable but also minimized the risk of errors.

We also explored alternative approaches to solving the problem, highlighting the flexibility and adaptability that are essential for mathematical proficiency. By regrouping terms and simplifying within brackets, we discovered a different pathway to the same solution. This reinforces the idea that there can be multiple ways to solve a mathematical problem, and the ability to identify the most efficient approach is a valuable skill.

Furthermore, we emphasized the importance of recognizing prime factorizations. By breaking down numbers into their prime factors, we gained a deeper understanding of their relationships and identified further opportunities for simplification. This technique is particularly useful for complex expressions with multiple terms and large numbers.

The final result, 4096 / 214358881, represents the simplified fractional form of the original expression. This solution demonstrates the power of mathematical simplification in reducing complex expressions to their most basic forms. The process of arriving at this solution has not only strengthened our understanding of mathematical operations but also instilled an appreciation for the beauty and elegance of mathematical reasoning.

Ultimately, the ability to simplify mathematical expressions is a valuable skill that extends far beyond the classroom. It is a fundamental tool for problem-solving in various fields, from science and engineering to finance and economics. By mastering these techniques, we empower ourselves to tackle complex challenges and make informed decisions in a data-driven world. This article serves as a reminder that mathematics is not just a collection of formulas and equations, but a powerful language for understanding and shaping the world around us.