Simplifying Expressions With Exponents: A Step-by-Step Guide

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Hey guys! Let's dive into the world of exponents and learn how to simplify expressions. Today, we're tackling an interesting problem: expressing the expression y^{1.7} ullet y^{2.8} ullet y^{-1.5} as a single power. Don't worry if it looks intimidating at first; we'll break it down step by step, making it super easy to understand. So, grab your thinking caps, and let's get started!

Understanding the Basics of Exponents

Before we jump into the problem, let's quickly recap the basics of exponents. An exponent tells us how many times to multiply a base by itself. For example, in the expression xnx^n, 'x' is the base, and 'n' is the exponent. This means we multiply 'x' by itself 'n' times. Understanding this fundamental concept is crucial for simplifying more complex expressions.

When dealing with expressions involving the same base raised to different powers, we can use some key exponent rules to simplify them. One of the most important rules, and the one we'll use today, is the product of powers rule. This rule states that when you multiply powers with the same base, you add the exponents. Mathematically, it looks like this: x^m ullet x^n = x^{m+n}. This rule is a game-changer when it comes to simplifying expressions like the one we're tackling. It allows us to combine multiple terms into a single, more manageable term. Another important rule to keep in mind is how to handle negative exponents. A negative exponent indicates that the base should be raised to the positive version of that exponent in the denominator. In other words, x−n=1xnx^{-n} = \frac{1}{x^n}. This rule is super helpful for dealing with terms like y−1.5y^{-1.5} in our expression. Remember, mastering these exponent rules is like having a superpower in algebra – it makes simplifying expressions a breeze!

Applying the Product of Powers Rule

Okay, now that we've refreshed our memory on exponent rules, let's get back to our problem: y^{1.7} ullet y^{2.8} ullet y^{-1.5}. The key here is to recognize that all three terms have the same base, which is 'y'. This is perfect because it means we can directly apply the product of powers rule we just discussed. According to this rule, when we multiply powers with the same base, we simply add the exponents.

So, let's add the exponents together: 1.7 + 2.8 + (-1.5). Notice that we're adding a negative number, which is the same as subtracting. This is a common situation when dealing with exponents, so it's good to be comfortable with it. Now, let's do the math. First, we can add 1.7 and 2.8, which gives us 4.5. Then, we subtract 1.5 from 4.5, which leaves us with 3. So, the sum of the exponents is 3. This means that when we combine the terms, the exponent of 'y' will be 3. Therefore, we can rewrite the original expression as y3y^3. See how simple that was? By applying the product of powers rule, we've successfully expressed the original expression as a single power of 'y'. This is a classic example of how exponent rules can significantly simplify algebraic expressions, making them much easier to work with. Remember this process; it's a fundamental skill in algebra!

Step-by-Step Solution

Let's break down the solution into a clear, step-by-step format so you can easily follow along and apply this method to similar problems.

  1. Identify the common base: In our expression, y^{1.7} ullet y^{2.8} ullet y^{-1.5}, the common base is 'y'. This is the foundation for applying the product of powers rule.
  2. Apply the product of powers rule: This rule states that x^m ullet x^n = x^{m+n}. We'll use this to combine the terms in our expression.
  3. Add the exponents: Add the exponents together: 1. 7 + 2.8 + (-1.5). Remember to pay attention to the signs, especially when dealing with negative exponents.
  4. Calculate the sum: 1. 7 + 2.8 = 4.5, and then 4.5 + (-1.5) = 3. So, the sum of the exponents is 3.
  5. Write the simplified expression: Now that we have the sum of the exponents, we can write the expression as a single power: y3y^3.

And that's it! We've successfully expressed the original expression as y3y^3 by following these simple steps. Breaking down the problem like this makes it much more manageable and helps to avoid errors. This step-by-step approach is not only useful for this specific problem but also for tackling any algebraic expression involving exponents. Make sure to practice these steps with different examples to solidify your understanding.

Common Mistakes to Avoid

When working with exponents, it's easy to make mistakes if you're not careful. Let's go over some common pitfalls to avoid, so you can ace your algebra problems every time.

One frequent mistake is forgetting the product of powers rule and incorrectly multiplying the exponents instead of adding them. Remember, the rule states that x^m ullet x^n = x^{m+n}, so you should always add the exponents when multiplying terms with the same base. Another common error is mishandling negative exponents. It's crucial to remember that a negative exponent means taking the reciprocal of the base raised to the positive exponent. For instance, x−n=1xnx^{-n} = \frac{1}{x^n}. Failing to apply this correctly can lead to significant errors in your calculations. Also, be cautious with the order of operations, especially when dealing with expressions involving multiple operations. Always follow the PEMDAS/BODMAS rule (Parentheses/Brackets, Exponents, Multiplication and Division, Addition and Subtraction) to ensure you're performing the operations in the correct sequence. Finally, double-checking your work is always a good practice. Exponent problems can sometimes involve tricky calculations, so taking a few extra moments to review your steps can help you catch any errors before they become a problem. By being aware of these common mistakes and taking steps to avoid them, you'll be well on your way to mastering exponents and simplifying algebraic expressions like a pro! Keep practicing, and you'll get the hang of it!

Practice Problems

To really master simplifying expressions with exponents, practice is key! Let's try a few more problems similar to the one we just solved. This will help you solidify your understanding and build confidence in your skills.

Here are a couple of practice problems for you to try:

  1. Simplify the expression: z^{2.5} ullet z^{1.5} ullet z^{-0.5}
  2. Express the following as a single power: a^{3.2} ullet a^{-1.2} ullet a^{0.8}

For the first problem, z^{2.5} ullet z^{1.5} ullet z^{-0.5}, follow the same steps we used earlier. Identify the common base (which is 'z' in this case), apply the product of powers rule, add the exponents (2.5 + 1.5 + (-0.5)), and then write the simplified expression. You should find that the simplified expression is z3.5z^{3.5}.

For the second problem, a^{3.2} ullet a^{-1.2} ullet a^{0.8}, again identify the common base ('a'), apply the product of powers rule, and add the exponents (3.2 + (-1.2) + 0.8). In this case, the sum of the exponents is 2.8, so the simplified expression is a2.8a^{2.8}.

Working through these practice problems will help you become more comfortable with the process and develop your problem-solving skills. Remember, the more you practice, the easier these concepts will become. So, keep at it, and you'll be simplifying expressions with exponents like a total expert in no time!

Conclusion

Alright, guys, we've covered a lot in this guide! We've learned how to simplify expressions with exponents by applying the product of powers rule. We tackled the problem y^{1.7} ullet y^{2.8} ullet y^{-1.5}, broke it down step by step, and found that it simplifies to y3y^3. We also discussed common mistakes to avoid and worked through some practice problems to solidify our understanding. Remember, the key to mastering exponents is understanding the rules and practicing consistently.

Simplifying expressions with exponents is a fundamental skill in algebra, and it's super useful for solving more complex equations and problems. So, make sure you've got a good grasp of these concepts. If you ever get stuck, don't hesitate to review this guide or ask for help. Keep practicing, and you'll become an exponent whiz in no time! And with that, we wrap up this guide. Keep up the awesome work, and happy simplifying! You've got this!