Evaluating The Expression M = [(-27)^(-2/3) + (-27)^(-5/3) + 2(3)^(-4)] A Step-by-Step Solution
Evaluate the expression M = [(-27)^(-2/3) + (-27)^(-5/3) + 2(3)^(-4)].
Introduction: Decoding the Mathematical Expression
In the realm of mathematics, expressions often present themselves as puzzles, challenging us to unravel their intricate layers and arrive at a concise solution. Today, we embark on a journey to evaluate the expression M = [(-27)^(-2/3) + (-27)^(-5/3) + 2(3)^(-4)]. This expression, seemingly complex at first glance, can be simplified by applying the fundamental principles of exponents and radicals. Our goal is to break down each term, understand its components, and ultimately combine them to find the value of M. This process involves navigating negative exponents, fractional exponents, and basic arithmetic operations. By the end of this exploration, you'll not only understand the solution but also gain a deeper appreciation for the elegance and precision of mathematical manipulations. So, let's delve into the world of exponents and unravel the mystery of this expression, step by step, ensuring a clear and comprehensive understanding along the way. The key to success lies in recognizing the properties of exponents and radicals and applying them judiciously to simplify each term before combining them. This meticulous approach will lead us to the final, simplified value of M, providing a satisfying conclusion to our mathematical endeavor. Remember, every complex expression is simply a combination of smaller, manageable parts, and by tackling each part systematically, we can conquer the whole. This is the essence of mathematical problem-solving, and it's the spirit we'll embrace as we embark on this journey.
Breaking Down the Expression: A Step-by-Step Approach
To effectively evaluate this expression, we'll adopt a step-by-step approach, dissecting each term and simplifying it individually before combining them. This methodical approach will not only make the process easier to follow but also minimize the chances of error. First, let's focus on the term (-27)^(-2/3). This term involves a negative base, a negative exponent, and a fractional exponent. To tackle this, we'll first address the negative exponent by taking the reciprocal of the base raised to the positive exponent. This gives us 1/((-27)^(2/3)). Now, we need to deal with the fractional exponent. Recall that a fractional exponent of the form a^(m/n) can be interpreted as the nth root of a raised to the power of m, or (n√a)^m. Applying this to our term, we get 1/((3√-27)^2). The cube root of -27 is -3, so the expression becomes 1/((-3)^2), which simplifies to 1/9. Now, let's move on to the second term, (-27)^(-5/3). Similar to the previous term, we'll first address the negative exponent by taking the reciprocal, resulting in 1/((-27)^(5/3)). Applying the fractional exponent rule, we get 1/((3√-27)^5). Again, the cube root of -27 is -3, so the expression becomes 1/((-3)^5). Since (-3)^5 equals -243, the term simplifies to -1/243. Finally, let's consider the third term, 2(3)^(-4). We address the negative exponent by taking the reciprocal, which gives us 2/(3^4). Since 3^4 equals 81, the term simplifies to 2/81. With each term simplified, we're now ready to combine them and find the final value of M. This step-by-step breakdown has transformed a seemingly daunting expression into a series of manageable calculations, highlighting the power of methodical problem-solving in mathematics.
Calculating the Value of M: Combining the Simplified Terms
With the individual terms of the expression simplified, we can now proceed to calculate the value of M by combining them. We have: (-27)^(-2/3) = 1/9, (-27)^(-5/3) = -1/243, and 2(3)^(-4) = 2/81. Therefore, M = 1/9 - 1/243 + 2/81. To add or subtract fractions, we need a common denominator. The least common multiple of 9, 243, and 81 is 243. So, we'll convert each fraction to have a denominator of 243. 1/9 is equivalent to 27/243. -1/243 remains as -1/243. 2/81 is equivalent to 6/243. Now we can rewrite the expression for M as: M = 27/243 - 1/243 + 6/243. Combining the numerators, we get: M = (27 - 1 + 6) / 243. This simplifies to: M = 32/243. Thus, the value of the expression M is 32/243. This final calculation brings our journey to a successful conclusion. By meticulously breaking down the original expression, simplifying each term individually, and then combining them with a common denominator, we have arrived at a concise and accurate answer. This process exemplifies the power of systematic problem-solving in mathematics, demonstrating how complex expressions can be tamed through the application of fundamental principles and careful execution. The result, 32/243, represents the culmination of our efforts, a testament to the elegance and precision of mathematical reasoning.
Conclusion: The Elegance of Mathematical Solutions
In conclusion, we have successfully evaluated the expression M = [(-27)^(-2/3) + (-27)^(-5/3) + 2(3)^(-4)], arriving at the solution M = 32/243. This journey through exponents, radicals, and fractions has underscored the importance of methodical problem-solving in mathematics. By breaking down the expression into manageable parts, simplifying each term individually, and then combining them with a common denominator, we were able to navigate the complexities and arrive at a clear and concise answer. This process not only demonstrates the power of mathematical tools and techniques but also highlights the beauty and elegance inherent in mathematical solutions. The initial expression, seemingly daunting, yielded to our systematic approach, revealing the underlying simplicity and harmony. The result, 32/243, stands as a testament to the precision and accuracy that mathematics demands and delivers. Moreover, this exercise reinforces the value of understanding fundamental mathematical principles, such as the properties of exponents and radicals, and applying them strategically to solve problems. Each step, from addressing negative exponents to finding common denominators, showcased the interconnectedness of mathematical concepts and the importance of a solid foundation. As we conclude this exploration, we carry with us not only the solution to this specific expression but also a deeper appreciation for the elegance and power of mathematics as a whole. The ability to break down complex problems, apply logical reasoning, and arrive at precise solutions is a valuable skill, both within the realm of mathematics and beyond.