Simplifying Exponential Expressions: A Step-by-Step Guide

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Hey everyone! Today, we're diving into the world of exponents and radicals. We're going to break down some expressions step-by-step to make sure you've got a solid grasp of how these work. Don't worry, it's not as scary as it looks. We'll be using some key properties of exponents to make things easier. Let's get started, shall we?

Understanding the Basics: Exponents and Roots

Before we jump into the expressions, let's quickly recap some fundamental concepts. An exponent tells us how many times to multiply a number by itself. For example, in the expression 2^3, the exponent is 3, and it means 2 * 2 * 2 = 8. Pretty straightforward, right? Now, what about fractional exponents? Fractional exponents, like the ones we'll see in the examples, are related to roots. The denominator of the fractional exponent represents the root to take. For instance, x^(1/2) is the same as the square root of x, and x^(1/3) is the cube root of x. Now, when you see a negative exponent, it means you need to take the reciprocal of the base raised to the positive value of the exponent. For example, x^(-2) is equal to 1 / x^2. These concepts are the building blocks that will help us solve the given expressions. Remember, practice makes perfect, so don’t hesitate to work through these examples multiple times until they become second nature. Understanding exponents and roots is not just about solving these specific problems; it's a fundamental skill that underpins more advanced mathematical concepts. So, let's keep our eyes on the prize and aim to fully grasp these basic operations, as they are crucial for solving a wide range of algebraic and calculus problems.

The Power of Properties

Remember the properties of exponents, guys. They're your best friends in this kind of problem. Things like the power of a power rule ( (xa)b = x^(a*b) ), the product of powers rule (x^a * x^b = x^(a+b) ), and the quotient of powers rule (x^a / x^b = x^(a-b) ) are super important. Understanding these properties will allow you to quickly simplify complicated expressions into something manageable. Make sure you're comfortable with these properties before you move on; it will make the whole process a lot easier! If you have any doubts, don't be afraid to revisit the basics, look up some extra examples, or ask for help. These properties are the tools of our trade when working with exponents, and mastering them is essential for becoming proficient in mathematics. Make sure you know them well; they are critical for understanding how to manipulate exponential expressions.

Let's Evaluate: Step-by-Step Solutions

Alright, let's solve those expressions! We'll go through each one slowly and explain every step. I'm sure you will learn something new today, so let's get into it, shall we?

Expression 1: (25(-3/2))(1/3) = ?

Okay, let's start with the first one: (25(-3/2))(1/3). This looks a bit intimidating at first, but we'll break it down. First, we need to deal with the inner exponent. Remember that a negative exponent means we take the reciprocal. So, 25^(-3/2) becomes 1 / 25^(3/2). Now, let's look at 25^(3/2). This can be interpreted as the square root of 25, which is 5, raised to the power of 3 (5^3). So, 5^3 equals 125. Thus, 25^(3/2) is 125. Now we have 1 / 125. The expression now is (1/125)^(1/3), which means the cube root of 1/125. The cube root of 1 is 1, and the cube root of 125 is 5. So, the answer is 1/5. This one is all about breaking the expression down into smaller, manageable chunks, and remembering the definition of fractional and negative exponents. By carefully applying the rules, we turned a complex-looking expression into a simple fraction. Now, we are ready to advance to the next expression, let’s go!

Detailed Breakdown

Let’s go through the steps again in detail.

  1. Original Expression: (25(-3/2))(1/3)
  2. Negative Exponent: Rewrite 25^(-3/2) as 1 / 25^(3/2)
  3. Fractional Exponent: Understand that 25^(3/2) is the same as (√25)^3, which is (5)^3
  4. Simplify: (5)^3 = 125
  5. Substitute: The expression becomes (1 / 125)^(1/3)
  6. Cube Root: Take the cube root of 1/125 which is 1/5

Therefore, (25(-3/2))(1/3) = 1/5. I hope this helps you understand the step by step process. Make sure you understand the basics before solving these complex questions.

Expression 2: (16^(5/4) * 16^(1/4)) / (16(1/2))2 = ?

Next up, we have (16^(5/4) * 16^(1/4)) / (16(1/2))2. First, let's simplify the numerator. Using the product of powers rule, we can add the exponents: 16^(5/4) * 16^(1/4) = 16^(5/4 + 1/4) = 16^(6/4), which simplifies to 16^(3/2). Now let’s simplify the denominator. We can apply the power of a power rule: (16(1/2))2 = 16^(1/2 * 2) = 16^1 = 16. So, we now have 16^(3/2) / 16. Now, let’s deal with the 16^(3/2) term. This means the square root of 16, which is 4, raised to the power of 3. So, 4^3 equals 64. Hence we now have 64 / 16. Finally, 64 divided by 16 is 4. Thus the solution to the expression is 4. Remember to pay close attention to each step, and don’t be afraid to take your time. This problem highlights how several properties of exponents can be combined to simplify a complex expression into a straightforward calculation. Practice is key, so try working through similar examples to boost your confidence and understanding of these concepts.

Detailed Breakdown

Let's break down the steps for the second expression:

  1. Original Expression: (16^(5/4) * 16^(1/4)) / (16(1/2))2
  2. Product of Powers: Simplify the numerator: 16^(5/4) * 16^(1/4) = 16^(5/4 + 1/4) = 16^(6/4) = 16^(3/2)
  3. Power of a Power: Simplify the denominator: (16(1/2))2 = 16^(1/2 * 2) = 16^1 = 16
  4. Fractional Exponent: Evaluate 16^(3/2): (√16)^3 = 4^3 = 64
  5. Division: Divide 64 / 16 = 4.

Therefore, (16^(5/4) * 16^(1/4)) / (16(1/2))2 = 4.

Expression 3: 27^2 / 27^(4/3) = ?

Alright, let’s tackle the last one: 27^2 / 27^(4/3). This is a good example of how the quotient of powers rule can be applied. When dividing exponential expressions with the same base, you subtract the exponents. So we have 27^(2 - 4/3). To subtract, we need a common denominator. We can rewrite 2 as 6/3. So we have 27^(6/3 - 4/3) = 27^(2/3). Now, we need to evaluate 27^(2/3). This means the cube root of 27 (which is 3) raised to the power of 2. So, 3^2 equals 9. Thus the solution to the last expression is 9. This problem emphasizes the usefulness of the quotient rule and the ease with which you can simplify exponential expressions by knowing your properties. Take your time, practice, and you will become proficient in handling these kinds of problems with confidence.

Detailed Breakdown

Let’s go through the steps for this expression:

  1. Original Expression: 27^2 / 27^(4/3)
  2. Quotient of Powers: Subtract the exponents: 27^(2 - 4/3)
  3. Common Denominator: Rewrite 2 as 6/3: 27^(6/3 - 4/3)
  4. Subtraction: Simplify the exponent: 27^(2/3)
  5. Fractional Exponent: Evaluate 27^(2/3): (∛27)^2 = 3^2 = 9

Therefore, 27^2 / 27^(4/3) = 9.

Conclusion: Mastering Exponents

And there you have it, guys! We've worked through three different exponential expressions, step-by-step. Remember that mastering these types of problems involves understanding the properties of exponents, breaking down complex expressions into simpler parts, and practicing consistently. Make sure you go back through these examples, try working through them on your own, and maybe even create your own practice problems. If you put in the time and effort, you'll find that exponents are not as hard as they seem. Keep practicing, and you'll be able to solve these with ease. Math is all about building skills one step at a time, so keep at it! Also, don't be afraid to use online resources, ask for help, or join study groups. The more you engage with the material, the better you will understand it. Keep up the excellent work! Now, go out there and conquer those exponents!