1. Solve For K In The Equation (9/49)^-5 × (9/49)^7 = (9/49)^-6k. 2. Solve For K In The Equation (1.4)^8 × (1.4)^5 = (1.4)^3 × (1.4)^k. 3. Solve For M In The Equation 5^m ÷ 5^-3 = 5^5. 4. Simplify Exponential Expressions.
Introduction
In the realm of mathematics, exponents play a pivotal role in expressing repeated multiplication and simplifying complex calculations. This article delves into the fascinating world of exponents, providing a comprehensive guide to solving exponential equations and simplifying expressions. We will explore fundamental concepts, unravel intricate problems, and equip you with the skills to conquer any exponential challenge. Our journey will encompass a series of carefully crafted examples, each designed to illuminate a specific aspect of exponential manipulation. We will begin by tackling equations involving fractional bases, gradually progressing to more complex scenarios that require a nuanced understanding of exponential properties. By the end of this exploration, you will possess a robust toolkit for navigating the intricacies of exponents and applying your knowledge to diverse mathematical contexts.
Problem 1: Cracking the Code of Fractional Exponents
Fractional exponents often appear daunting, but with a solid grasp of exponent rules, they become surprisingly manageable. Let's tackle our first challenge: If (9/49)^-5 × (9/49)^7 = (9/49)^-6k, find the value of k. This problem elegantly demonstrates the power of the product of powers rule, which states that when multiplying exponents with the same base, you simply add the powers. In this case, our base is the fraction 9/49, and the exponents are -5 and 7. Applying the rule, we can simplify the left side of the equation: (9/49)^-5 × (9/49)^7 = (9/49)^(-5 + 7) = (9/49)^2. Now our equation looks much simpler: (9/49)^2 = (9/49)^-6k. The next crucial step involves recognizing that if two exponential expressions with the same base are equal, then their exponents must also be equal. This principle allows us to transform the equation into a straightforward algebraic problem: 2 = -6k. Solving for k, we divide both sides by -6, yielding k = -1/3. Therefore, the value of k that satisfies the given equation is -1/3. This problem not only showcases the application of the product of powers rule but also highlights the fundamental connection between exponential equations and algebraic solutions. By mastering these principles, you'll be well-equipped to tackle a wide range of exponential challenges.
Problem 2: Mastering the Product of Powers Rule
The product of powers rule is a cornerstone of exponent manipulation, and our second problem provides an excellent opportunity to solidify your understanding. We are tasked with finding the value of k in the equation (1.4)^8 × (1.4)^5 = (1.4)^3 × (1.4)^k. This equation showcases the rule in action on both sides, requiring us to apply it strategically to isolate the unknown variable. Let's begin by simplifying the left side of the equation. Using the product of powers rule, we add the exponents 8 and 5: (1.4)^8 × (1.4)^5 = (1.4)^(8 + 5) = (1.4)^13. Now the equation looks like this: (1.4)^13 = (1.4)^3 × (1.4)^k. Next, we apply the same rule to the right side of the equation. We need to combine the terms with the base 1.4, but one of the exponents is the unknown k. So, we write: (1.4)^3 × (1.4)^k = (1.4)^(3 + k). Now our equation is further simplified: (1.4)^13 = (1.4)^(3 + k). As we learned in the previous problem, if two exponential expressions with the same base are equal, their exponents must be equal. This gives us the algebraic equation: 13 = 3 + k. To solve for k, we subtract 3 from both sides: k = 13 - 3 = 10. Therefore, the value of k that satisfies the equation is 10. This problem reinforces the importance of the product of powers rule and demonstrates how it can be used to solve equations with exponents on both sides. By breaking down the problem into smaller steps and applying the rule systematically, we can arrive at the solution with confidence. The ability to manipulate exponents in this way is crucial for tackling more advanced mathematical concepts.
Problem 3: Unveiling the Quotient of Powers Rule
The quotient of powers rule is another essential tool in our exponential arsenal. It governs the division of exponents with the same base, stating that you subtract the exponents. Let's see how this rule applies in the following problem: Find the value of m, such that 5^m ÷ 5^-3 = 5^5. This problem introduces a negative exponent, which might seem intimidating at first, but the quotient of powers rule handles it elegantly. Remember that dividing by a number with an exponent is the same as multiplying by the number with the negative of that exponent. So, 5^m ÷ 5^-3 can be rewritten using the quotient of powers rule as 5^(m - (-3)). This simplifies to 5^(m + 3). Now our equation is 5^(m + 3) = 5^5. As before, if two exponential expressions with the same base are equal, their exponents must be equal. This gives us the algebraic equation: m + 3 = 5. To solve for m, we subtract 3 from both sides: m = 5 - 3 = 2. Therefore, the value of m that satisfies the given condition is 2. This problem highlights the versatility of the quotient of powers rule, particularly when dealing with negative exponents. By understanding and applying this rule, you can confidently navigate expressions involving division of exponents. The combination of the product of powers rule and the quotient of powers rule provides a powerful framework for simplifying and solving a wide variety of exponential problems. These rules are fundamental to many areas of mathematics, and mastering them will significantly enhance your problem-solving abilities.
Simplify:
This section will focus on simplifying various exponential expressions, further honing your skills in applying exponent rules. Simplifying expressions is a crucial step in many mathematical problems, allowing you to reduce complexity and reveal underlying patterns. We will explore a range of examples, each designed to showcase a different aspect of exponent simplification. The key to success in this area lies in a thorough understanding of the exponent rules and the ability to apply them strategically. Remember, the goal is to reduce the expression to its simplest form, typically with the fewest possible terms and no negative exponents. We will tackle expressions involving products, quotients, powers of powers, and combinations of these operations. By working through these examples, you will develop a strong intuition for exponent manipulation and gain the confidence to tackle even the most challenging simplification problems. Mastering these techniques will not only enhance your mathematical skills but also provide a valuable foundation for more advanced topics such as algebra, calculus, and beyond. So, let's embark on this journey of simplification and unlock the hidden beauty within exponential expressions.
Conclusion
Throughout this exploration, we've journeyed through the core principles of exponents, tackling equations and simplifying expressions with confidence. We've dissected the product of powers rule, the quotient of powers rule, and the crucial connection between exponential and algebraic forms. Each problem served as a stepping stone, solidifying your grasp of these fundamental concepts. Remember, the key to success in mathematics lies not just in memorizing rules, but in understanding their application and practicing consistently. As you continue your mathematical journey, the knowledge and skills you've gained here will serve as a solid foundation for more advanced topics. Embrace the challenges that come your way, and never stop exploring the fascinating world of numbers and equations. The power of exponents, once mastered, will unlock new realms of mathematical understanding and empower you to solve problems with elegance and efficiency.