Simplifying Expressions With Exponents A Step-by-Step Guide
This article provides a detailed walkthrough of simplifying complex mathematical expressions involving exponents. We will dissect two intricate problems, breaking down each step to ensure a clear understanding of the underlying principles. Mastering these techniques is crucial for anyone delving into advanced mathematics, physics, or engineering.
Part 1: Simplifying the First Expression
The first expression we'll tackle is: (128)^{-2/7} - (625{-3}){-1/4} + 14(2401)^{-1/4}. This expression involves negative fractional exponents, which might seem daunting at first. However, by systematically applying the rules of exponents, we can simplify it effectively.
Step 1: Understanding Negative Exponents
The key to simplifying expressions with negative exponents lies in the rule: a^{-n} = 1/a^n. This rule states that a number raised to a negative exponent is equal to the reciprocal of that number raised to the positive exponent. Applying this rule to our expression, we get:
(128)^{-2/7} becomes 1/(128^{2/7}) (625{-3}){-1/4} remains as is for now, but we will address it in the next step. (2401)^{-1/4} becomes 1/(2401^{1/4})
Step 2: Simplifying Fractional Exponents
Fractional exponents represent both roots and powers. The denominator of the fraction indicates the root to be taken, and the numerator indicates the power to which the result should be raised. For example, a^{m/n} is the nth root of a raised to the mth power, or (a(1/n))m. Now, we will dive deeper into fractional exponents. A fractional exponent, such as a^(m/n), signifies a combination of exponentiation and root extraction. The denominator 'n' indicates the type of root to be taken (e.g., square root for n=2, cube root for n=3), while the numerator 'm' indicates the power to which the base should be raised. This concept can be mathematically expressed as: a^(m/n) = (n√a)^m = n√(a^m). Understanding this dual nature of fractional exponents is crucial for simplifying complex mathematical expressions. It allows us to transform exponents into radicals and vice versa, depending on which form simplifies the expression more easily. This flexibility is a powerful tool in manipulating and solving equations, making the simplification process more efficient and intuitive. For instance, in our current problem, recognizing the fractional exponents as both roots and powers enables us to break down each term into manageable parts, making it easier to identify perfect powers or roots. This step-by-step approach not only simplifies the calculations but also deepens the understanding of the underlying mathematical principles, fostering a stronger foundation in algebra and calculus. By mastering the intricacies of fractional exponents, one gains a significant advantage in tackling more advanced mathematical problems, enhancing problem-solving skills and analytical abilities.
Step 3: Evaluating 128^{2/7}
To evaluate 128^2/7}, we first find the 7th root of 128. We recognize that 128 is 2^7, so the 7th root of 128 is 2. Then, we square the result = 4.
Step 4: Simplifying (625{-3}){-1/4}
When raising a power to another power, we multiply the exponents. So, (625{-3})-1/4} becomes 625^{(-3)*(-1/4)} = 625^{3/4}. Now, we find the 4th root of 625. We know that 625 is 5^4, so the 4th root of 625 is 5. Then, we cube the result)^{-1/4} = 125.
Step 5: Evaluating 2401^{-1/4}
We already know that 2401^{-1/4} becomes 1/(2401^{1/4}). We need to find the 4th root of 2401. We recognize that 2401 is 7^4, so the 4th root of 2401 is 7. Therefore, 2401^{-1/4} = 1/7.
Step 6: Putting it All Together
Now we substitute the simplified terms back into the original expression:
1/(128^{2/7}) - (625{-3}){-1/4} + 14(2401)^{-1/4} becomes 1/4 - 125 + 14(1/7)
Simplifying further, we get:
1/4 - 125 + 2 = 1/4 - 123 = (1 - 492)/4 = -491/4
Therefore, the simplified form of the first expression is -491/4.
Part 2: Simplifying the Second Expression
The second expression is: 4/(2187^{-3/7}) - 5/(256^{-1/4}) + 2/(1331^{-1/3}). This expression involves multiple terms with negative fractional exponents in the denominators. We will again use the rules of exponents to simplify each term individually and then combine them.
Step 1: Dealing with Negative Exponents in Denominators
Recall that a^{-n} = 1/a^n. Therefore, 1/a^{-n} = a^n. This rule is essential for simplifying expressions with negative exponents in the denominator. Applying this to our expression, we get:
4/(2187^{-3/7}) becomes 4 * (2187^{3/7}) 5/(256^{-1/4}) becomes 5 * (256^{1/4}) 2/(1331^{-1/3}) becomes 2 * (1331^{1/3})
Step 2: Evaluating 2187^{3/7}
To evaluate 2187^3/7}, we first find the 7th root of 2187. We recognize that 2187 is 3^7, so the 7th root of 2187 is 3. Then, we cube the result = 27. Let's further delve into the significance of understanding the relationship between bases and their roots. In simplifying expressions with fractional exponents, identifying the base as a power of another number is a pivotal step. This skill streamlines the simplification process, enabling efficient extraction of roots and computation of powers. For instance, in the case of 2187^{3/7}, recognizing 2187 as 3^7 allows us to readily find the 7th root of 2187, which is 3. This recognition significantly simplifies the calculation. This ability to discern these relationships is not merely a computational shortcut; it demonstrates a deeper understanding of number theory and the properties of exponents. It enhances one's problem-solving acumen by allowing for quicker and more accurate simplification of complex expressions. Moreover, this foundational knowledge is invaluable in more advanced mathematical contexts, such as calculus and differential equations, where efficient manipulation of exponential and logarithmic functions is crucial. Developing this skill through practice and exposure to various numerical problems not only improves algebraic proficiency but also fosters a more intuitive and insightful approach to mathematical problem-solving.
Step 3: Evaluating 256^{1/4}
To evaluate 256^{1/4}, we need to find the 4th root of 256. We recognize that 256 is 4^4 (or 2^8), so the 4th root of 256 is 4. Therefore, 256^{1/4} = 4.
Step 4: Evaluating 1331^{1/3}
To evaluate 1331^{1/3}, we need to find the cube root of 1331. We know that 1331 is 11^3, so the cube root of 1331 is 11. Therefore, 1331^{1/3} = 11.
Step 5: Putting it All Together
Now we substitute the simplified terms back into the original expression:
4 * (2187^{3/7}) - 5 * (256^{1/4}) + 2 * (1331^{1/3}) becomes 4 * 27 - 5 * 4 + 2 * 11
Simplifying further, we get:
108 - 20 + 22 = 110
Therefore, the simplified form of the second expression is 110.
Conclusion
Simplifying complex expressions with exponents requires a solid understanding of the rules of exponents and the ability to recognize powers and roots. By breaking down each expression into smaller, manageable steps, we can effectively simplify even the most challenging problems. Mastering these techniques is essential for success in advanced mathematical studies and various scientific fields. This comprehensive guide has walked through the step-by-step simplification of two intricate expressions, emphasizing the importance of understanding fractional and negative exponents, and the ability to identify perfect powers and roots. These skills are not only crucial for academic success but also for problem-solving in real-world applications across various disciplines. By consistently practicing and applying these techniques, students and professionals alike can enhance their mathematical proficiency and analytical capabilities, leading to more effective and efficient problem-solving in any context. Moreover, a deep understanding of these concepts forms a solid foundation for further exploration in advanced mathematics, physics, engineering, and computer science, where exponential functions and their properties are frequently encountered. Thus, the effort invested in mastering these fundamental principles yields long-term benefits, empowering individuals to tackle complex challenges with confidence and precision.
