Simplifying Algebraic Expressions: A Step-by-Step Guide

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Hey guys! Let's dive into a fun math problem. We're gonna break down the expression: (-x²)³ •(-x³)²• (x)-⁸. Don't worry if it looks a bit intimidating at first; we'll take it slow and steady. The goal here is to simplify this expression using the rules of exponents. This involves understanding how to handle powers of powers, products of powers, and negative exponents. We'll also pay close attention to the signs, since they can easily trip you up if you're not careful. Remember, math is all about following the rules – once you know them, you can solve just about anything! By the end of this explanation, you'll not only know the answer but also understand the reasoning behind each step. So, grab a pen and paper, and let's get started! We're going to rewrite and simplify the expression step by step, making sure you grasp every concept along the way. The key to success in these types of problems is to break them down into smaller, more manageable parts. This approach prevents errors and builds your understanding in a structured manner. We’ll be using the properties of exponents: the power of a power rule, the product of powers rule, and the negative exponent rule. We’ll also be very careful with the negative signs, ensuring that we understand how they affect the final outcome. This problem gives us a chance to refresh our skills and boost our confidence with algebra! This will enhance your skills and make solving complex problems seem like a breeze. So, let's get started and get this thing solved together. We'll make sure you're well-equipped to tackle similar problems in the future. It's all about practice, consistency, and understanding the underlying principles of mathematics!

Breaking Down the Problem: The Power of a Power Rule

First up, let's tackle the (-x²)³ part of our expression. Remember the power of a power rule? It states that when you have an exponent raised to another exponent, you multiply the exponents. So, (x^m)^n = x^(m*n). Applying this to our first term, we have (-x²)³ = (-1)³ * (x²) ³.

  • Here, the (-1)³ indicates that the negative sign is also affected by the power. Since the exponent is odd, the result will remain negative. This gives us -1 * x^(2*3). When we multiply the exponents, we get x^(2*3) = x^6. Thus, (-x²)³ simplifies to -x⁶. It’s essential to pay attention to both the sign and the exponents. This step highlights how the power affects every part of the base, including the negative sign, which can often be missed. Taking the time to break down each term like this helps avoid simple calculation errors. It also allows us to clearly see how the rules of exponents are applied, providing a solid foundation for more complex problems.

Next, let's simplify (-x³)². Applying the power of a power rule once more, we get (-x³)² = (-1)² * (x³)². Here, because the exponent is even, the negative sign disappears, leaving us with just a positive one, then multiplying exponents again gives x^(3*2) = x^6. Thus (-x³)² simplifies to x⁶. Notice the difference from the previous step, this underscores the importance of considering the sign and how it's affected by the exponent. This attention to detail reduces the likelihood of making common mistakes, and builds a deeper understanding of the rules. This step is a good example of how careful consideration of the signs can directly impact the overall result of the algebraic expression.

Combining the Terms: The Product of Powers Rule

Now, we have the simplified parts: -x⁶ from (-x²)³ and x⁶ from (-x³)². Let's put these back into our original expression and also bring in the last term (x)-⁸ that we did not touch yet. So far, we have -x⁶ * x⁶ * x⁻⁸. Remember the product of powers rule? When multiplying powers with the same base, you add the exponents: x^m * x^n = x^(m+n). First, let's tackle the product of the x terms: x⁶ * x⁶ * x⁻⁸. By applying the product of powers rule, we add the exponents: 6 + 6 + (-8) = 4. Thus we have x⁴. Now, our expression looks like -1 * x⁴ or just -x⁴.

Remember that (-1) multiplies the result. This is a crucial step. The main thing to remember is the multiplication of the result by the negative sign that appears as a result of the first term. That means we are multiplying our final result by the negative sign. This step is very important because it clearly shows the importance of handling the negative signs correctly. By keeping the (-1) separate, we ensure that it is applied correctly to the final result. Doing this helps us stay organized and avoid errors. Combining terms is a common operation in algebra, so mastering this is key to handling more complex expressions.

The Final Answer: Simplifying to the Core

Finally, let's put it all together. We have reduced our original expression to -x⁴. This is the final simplified form of the expression (-x²)³ •(-x³)²• (x)-⁸. Therefore, the result of the operation (-x²)³ •(-x³)²• (x)-⁸ is -x⁴. This final answer is a testament to the power of step-by-step simplification. We've successfully broken down a complex expression into manageable parts, applied the correct rules of exponents, and handled signs carefully. This process shows how to simplify an algebraic expression and also highlights the importance of precision in mathematics. Knowing how to correctly apply these principles builds a strong foundation for more advanced algebraic concepts. Always make sure you understand each step. If you're ever unsure, don't hesitate to review the rules of exponents and practice with similar problems. Keep practicing and you'll become a pro at simplifying these types of expressions!

In summary, here's how we got there:

  1. Apply the power of a power rule to (-x²)³ to get -x⁶.
  2. Apply the power of a power rule to (-x³)² to get x⁶.
  3. Rewrite the original expression as -x⁶ * x⁶ * x⁻⁸.
  4. Apply the product of powers rule to simplify x⁶ * x⁶ * x⁻⁸ to x⁴.
  5. Combine all terms to obtain the final answer -x⁴.

Congratulations, guys! You've successfully simplified a complex algebraic expression! Keep practicing, and you'll be aceing these problems in no time. Always remember, math is a skill that improves with practice. Don’t give up, keep going and you will become masters. Now you're one step closer to algebraic mastery. Well done!