Solving Exponential Expressions: A Step-by-Step Guide

Hey guys! Today, we're diving into the exciting world of exponential expressions. We'll be tackling two problems that involve exponents and basic arithmetic operations. Don't worry, it's not as scary as it sounds! We'll break it down step-by-step, so you can follow along and conquer these mathematical challenges. Let's get started!

a) 3^(-3) * 2^5: Unraveling Negative and Positive Exponents

This first problem, 3^(-3) * 2^5, presents us with a combination of a negative exponent and a positive exponent. To solve it effectively, we need to understand how each type of exponent works. First and foremost, in tackling mathematical expressions, particularly those involving exponents, it’s crucial to adhere to the fundamental order of operations. This principle ensures consistency and accuracy in calculations. The order, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which operations must be performed. We'll start by addressing the exponents before moving on to multiplication. So, put on your thinking caps, and let's dive into the details!

Understanding Negative Exponents

The key thing to remember about negative exponents is that they represent reciprocals. A number raised to a negative power is the same as 1 divided by that number raised to the positive version of that power. In simpler terms, x^(-n) is the same as 1 / x^n. Applying this to our problem, 3^(-3) can be rewritten as 1 / 3^3. This transformation is crucial because it allows us to work with positive exponents, which are generally easier to handle. The negative exponent indicates that the base (in this case, 3) should be placed in the denominator of a fraction, with 1 as the numerator. This understanding is fundamental to simplifying expressions involving negative exponents.

Let's calculate 3^3. This means 3 multiplied by itself three times: 3 * 3 * 3. So, 3 * 3 equals 9, and then 9 * 3 equals 27. Therefore, 3^3 is 27. Now, we can substitute this value back into our expression. 1 / 3^3 becomes 1 / 27. Remember, understanding this transformation is key to working with negative exponents. It's not just about flipping the sign; it's about understanding the reciprocal relationship.

Tackling Positive Exponents

Now, let's look at the positive exponent in our expression: 2^5. This means 2 multiplied by itself five times: 2 * 2 * 2 * 2 * 2. Let's break it down step by step. 2 * 2 equals 4. Then, 4 * 2 equals 8. Next, 8 * 2 equals 16. Finally, 16 * 2 equals 32. So, 2^5 is 32. Calculating positive exponents involves straightforward multiplication, making it easier to grasp once the concept of repeated multiplication is understood. This component of the problem is more direct, but it's just as important for arriving at the final solution.

Putting It All Together

Now that we've calculated both 3^(-3) and 2^5, we can substitute these values back into our original expression. We have (1 / 27) * 32. To multiply a fraction by a whole number, we can think of the whole number as a fraction with a denominator of 1. So, we have (1 / 27) * (32 / 1). When multiplying fractions, we multiply the numerators (the top numbers) and the denominators (the bottom numbers). Therefore, we have (1 * 32) / (27 * 1), which simplifies to 32 / 27. This fraction is our final answer in its simplest form, as 32 and 27 have no common factors other than 1. So, the solution to 3^(-3) * 2^5 is 32/27. This result demonstrates the power of understanding and applying exponent rules, even when dealing with both negative and positive powers within the same expression.

b) 6^3 + (-5)^2: Combining Exponents and Addition

Our second problem, 6^3 + (-5)^2, involves calculating exponents and then adding the results. The key here is to remember the order of operations (PEMDAS/BODMAS) – exponents come before addition. We'll calculate each exponential term separately and then add them together. Let's break it down step-by-step to make sure we get the right answer. This type of problem highlights the importance of understanding how exponents interact with different arithmetic operations, especially when negative numbers are involved. So, let's roll up our sleeves and get into the nitty-gritty details!

Calculating 6^3

First, let's calculate 6^3. This means 6 multiplied by itself three times: 6 * 6 * 6. So, let's do the math: 6 * 6 equals 36. Now, we multiply 36 by 6. 36 * 6 equals 216. Therefore, 6^3 is 216. This calculation is a straightforward application of the exponent, representing repeated multiplication. It's important to perform this calculation accurately, as it forms the basis for the next step in the problem. Understanding the concept of exponents as repeated multiplication is fundamental here.

Dealing with (-5)^2

Next, we need to calculate (-5)^2. This means -5 multiplied by itself: (-5) * (-5). Remember the rules of multiplying negative numbers: a negative number multiplied by a negative number results in a positive number. So, -5 * -5 equals 25. Therefore, (-5)^2 is 25. This is a crucial point to remember: squaring a negative number always results in a positive number. This rule is essential for solving problems involving exponents and negative bases, as it directly impacts the sign of the result. Understanding this principle helps avoid common errors and ensures accurate calculations.

Adding the Results

Now that we've calculated both 6^3 and (-5)^2, we can add the results together. We have 216 + 25. Adding these two numbers is a simple arithmetic operation. 216 plus 25 equals 241. Therefore, the final answer to 6^3 + (-5)^2 is 241. This step is straightforward but crucial for completing the problem. It demonstrates the importance of following the order of operations and accurately combining the results of individual calculations. With this final addition, we've successfully solved the problem, showcasing our ability to work with exponents, negative numbers, and addition.

Conclusion: Mastering Exponential Expressions

So, there you have it, guys! We've successfully solved two problems involving exponential expressions. Remember, the key to mastering these types of problems is to understand the rules of exponents, especially negative exponents, and to follow the order of operations. Practice makes perfect, so keep working on these types of problems, and you'll become a pro in no time! Understanding the nuances of mathematical expressions like these not only builds a strong foundation in algebra but also sharpens your problem-solving skills, which are valuable in various fields. By breaking down complex problems into manageable steps, as we've done here, you can tackle any mathematical challenge with confidence. Keep exploring, keep practicing, and most importantly, keep enjoying the world of mathematics! Remember, each problem solved is a step closer to mastering the art of calculation and logical thinking. You've got this!