Simplifying Algebraic Expressions With Positive Exponents

Hey math enthusiasts! Let's dive into the world of simplifying algebraic expressions. This particular problem, $4w^2x^{-7}y^2 \cdot 4x^6w \cdot 4y^{-1}$, might look a bit intimidating at first glance, but trust me, it's totally manageable! We're going to break it down step by step, ensuring that our final answer uses only positive exponents. This is a fundamental skill in algebra, and mastering it will make your future math endeavors much smoother. Think of it like learning the alphabet before you write a novel – essential! We will learn how to approach the problem, the rules we need to remember, and how to arrive at the solution. So, grab your pencils, and let's get started. We will learn how to conquer expressions that involve exponents, variables, and coefficients. This skill is critical for more advanced topics. The process may seem intricate at first, but with practice, it becomes second nature. Each step builds upon the previous one. We will explore how to group similar terms, apply exponent rules, and simplify the expression to its most compact form. Simplifying expressions is not just about getting the right answer. It's about developing a deeper understanding of algebraic principles. This will help you to solve more complex equations. It will sharpen your critical thinking and problem-solving skills. Remember that this process will help you in your future career.

Before we begin, let's briefly recap some essential exponent rules. First, when multiplying terms with the same base, we add the exponents. For example, $x^m \cdot x^n = x^{m+n}$. Second, any term raised to a negative exponent can be rewritten as a fraction with a positive exponent. For instance, $x^{-n} = \frac{1}{x^n}$. Knowing these rules will be instrumental in solving the expression. Mastering these rules will greatly improve your problem-solving abilities. Ready to simplify the expression and eliminate those negative exponents? Let's do it! We will see how combining similar terms simplifies the overall expression. These concepts lay the foundation for solving more complicated equations. We aim to transform this seemingly complex expression into a simpler one.

Let's get down to the basics. Remember the goal: to simplify the expression $4w^2x^{-7}y^2 \cdot 4x^6w \cdot 4y^{-1}$ and express the answer with positive exponents only. Keep in mind that understanding is the key.

Step 1: Grouping Similar Terms and Multiplying Coefficients

Alright, guys, our first move is to group similar terms together. This means we'll gather the coefficients (the numbers), the w terms, the x terms, and the y terms separately. The original expression is $4w^2x^{-7}y^2 \cdot 4x^6w \cdot 4y^{-1}$. We begin by multiplying the coefficients: $4 \cdot 4 \cdot 4 = 64$. Now, let's group the variables. We have $w^2$ and $w$, which when multiplied together becomes $w^3$ (remember, $w$ is the same as $w^1$, and we add the exponents: $2 + 1 = 3$). For the x terms, we have $x^{-7}$ and $x^6$. For the y terms, we have $y^2$ and $y^{-1}$.

This gives us $64 \cdot w^3 \cdot x^{-7} \cdot x^6 \cdot y^2 \cdot y^{-1}$. Grouping similar terms helps to organize the expression. The initial grouping will make it easier to apply the exponent rules. It is an essential step towards simplification. This step helps to reduce confusion and make the next steps easier. Always remember this initial step. We will now apply the exponent rules.

Now we have $64w^3x^{-7}x^6y^2y^{-1}$. The next step will be to deal with the exponents, remembering our goal is to eliminate the negative exponents.

Step 2: Applying the Exponent Rules

Now comes the fun part: applying those exponent rules we talked about earlier. Let's tackle the x terms first. We have $x^{-7} \cdot x^6$. When multiplying terms with the same base, we add the exponents. So, we get $x^{-7+6} = x^{-1}$. Then, we'll deal with the y terms: $y^2 \cdot y^{-1} = y^{2-1} = y^1$ or simply $y$.

So far, our expression looks like this: $64w^3x^{-1}y$. We're almost there! Notice that we still have a negative exponent in $x^{-1}$. We need to convert it to a positive exponent. The key is to remember that $x^{-n} = \frac{1}{x^n}$.

Let's break down the x terms. We used the rule that when multiplying terms with the same base, you add the exponents. If the base is x, then $x^{-7} \cdot x^6 = x^{-7+6} = x^{-1}$. For the y terms, we have $y^2 \cdot y^{-1} = y^{2+(-1)} = y^{2-1} = y^1$, which is just y. So, we now have $64w^3x^{-1}y$. To get rid of that negative exponent, we use the rule $x^{-n} = \frac{1}{x^n}$.

Step 3: Eliminating Negative Exponents and Final Simplification

Alright, it's time to deal with that negative exponent and rewrite the expression with only positive exponents. We have $x^{-1}$, which, according to our rule, is equal to $\frac{1}{x}$. So, we replace $x^{-1}$ with $\frac{1}{x}$.

This transforms our expression from $64w^3x^{-1}y$ to $64w^3 \cdot \frac{1}{x} \cdot y$. Now we can rewrite this by combining the terms: $\frac{64w^3y}{x}$.

And that's it, guys! We have successfully simplified the expression $4w^2x^{-7}y^2 \cdot 4x^6w \cdot 4y^{-1}$ to $\frac{64w^3y}{x}$. This final form has only positive exponents. Congratulations on completing the simplification process. Remember that the process includes grouping, applying the exponent rules, and rewriting any negative exponents as fractions. This process is applicable to many algebraic expressions. This example showcases the power of systematic problem-solving. It's a key skill in mathematics and beyond! Remember that practice makes perfect, and with each problem you solve, your understanding deepens. The process isn't that hard, right?

So, to recap, our final answer is $\frac{64w^3y}{x}$. It is very important to have an understanding of the concepts. Keep practicing!

Conclusion: Mastering Algebraic Simplification

We have successfully simplified the algebraic expression $4w^2x^{-7}y^2 \cdot 4x^6w \cdot 4y^{-1}$! We took it step by step, applying exponent rules and ensuring our final answer used only positive exponents. Remember, the journey from the initial expression to the simplified form is a testament to the power of these concepts. Each step is essential. We have gained experience and enhanced our abilities. Always be prepared to apply these principles. The skills will be valuable in more advanced mathematical topics.

By simplifying, we have transformed a complex expression into a more manageable form. This process not only reveals the structure of the expression but also strengthens our understanding. Remember that the journey of learning math is similar to the journey of simplifying algebraic expressions. We start with complex problems and break them down. By following the steps in this guide, you have not only arrived at the correct answer but also improved your critical-thinking skills. These skills will serve you well in various areas. Keep practicing, keep learning, and keep simplifying! Congratulations on your hard work. You're now one step closer to algebraic mastery! Keep practicing, and you'll find that simplifying algebraic expressions becomes second nature. And remember, math can be fun! Go out there and apply these skills to conquer new problems. Now, you are ready to face any algebraic challenge. With consistent practice, you'll become proficient in no time. You can do it!