Simplify $2 X^{-1} imes 3 Y^2 imes 6 W^5 W^{-2} X Y^6$ With Positive Exponents

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Simplifying algebraic expressions is a fundamental skill in mathematics, often encountered in algebra and beyond. This article delves into simplifying the expression $2 x^{-1} imes 3 y^2 imes 6 w^5 w^{-2} x y^6$, providing a step-by-step guide to ensure clarity and understanding. Our primary goal is to manipulate the expression so that all exponents are positive, adhering to standard mathematical conventions and making the result more readable and usable. This involves combining like terms, applying the rules of exponents, and handling negative exponents appropriately.

Understanding the Basics of Exponents

Before diving into the simplification process, it’s crucial to understand the basic rules of exponents. These rules form the bedrock of our simplification strategy, allowing us to manipulate expressions involving powers efficiently. The key rules we will utilize include:

1. Product of Powers Rule: When multiplying like bases, add the exponents: $a^m imes a^n = a^{m+n}$. This rule is fundamental in combining terms with the same variable but different exponents.
2. Quotient of Powers Rule: When dividing like bases, subtract the exponents: $a^m / a^n = a^{m-n}$. Although not directly used in this expression, it is a related concept useful in other simplifications.
3. Power of a Power Rule: When raising a power to another power, multiply the exponents: $(a^m)^n = a^{mn}$. This rule is essential when dealing with expressions enclosed in parentheses and raised to a power.
4. Negative Exponent Rule: A term with a negative exponent can be rewritten as its reciprocal with a positive exponent: $a^{-n} = 1/a^n$. This is particularly important for our goal of expressing the final answer with only positive exponents.
5. Zero Exponent Rule: Any non-zero number raised to the power of zero is 1: $a^0 = 1$. This rule simplifies expressions where exponents might cancel out to zero.

These rules collectively allow us to simplify complex expressions by combining like terms and adjusting exponents. In the context of our given expression, we'll primarily use the product of powers rule and the negative exponent rule to achieve the desired simplification.

Step-by-Step Simplification

To effectively simplify the expression $2 x^{-1} imes 3 y^2 imes 6 w^5 w^{-2} x y^6$, we will proceed through a series of steps, each designed to clarify and organize the terms. The goal is to combine like terms, apply the rules of exponents, and ensure that all exponents are positive. Let’s break down the process:

Step 1: Group Like Terms

The first step in simplifying the expression is to group together the like terms. Like terms are those that have the same variable raised to potentially different powers. This grouping helps in visualizing and combining the coefficients and variables more clearly. In our expression, we have numerical coefficients, terms with the variable $x$, terms with the variable $y$, and terms with the variable $w$. Grouping these gives us:

$(2 imes 3 imes 6) imes (x^{-1} imes x) imes (y^2 imes y^6) imes (w^5 imes w^{-2})$

This arrangement makes it easier to see which terms can be combined using the rules of exponents.

Step 2: Multiply Coefficients

Next, we multiply the numerical coefficients together. This is a straightforward arithmetic operation:

$2 imes 3 imes 6 = 36$

So, the coefficient part of our expression simplifies to 36. Now we can focus on the variable terms.

Step 3: Combine $x$ Terms

We have $x^{-1}$ and $x$ terms to combine. Recall the product of powers rule: $a^m imes a^n = a^{m+n}$. Applying this rule to our $x$ terms, we get:

$x^{-1} imes x = x^{-1} imes x^1 = x^{-1+1} = x^0$

Any non-zero number raised to the power of zero is 1, so:

$x^0 = 1$

This simplifies the $x$ part of the expression to 1, effectively eliminating $x$ from this part of the expression.

Step 4: Combine $y$ Terms

Now, let's combine the $y$ terms: $y^2$ and $y^6$. Again, using the product of powers rule:

$y^2 imes y^6 = y^{2+6} = y^8$

So, the $y$ part of the expression simplifies to $y^8$.

Step 5: Combine $w$ Terms

Next, we combine the $w$ terms: $w^5$ and $w^{-2}$. Applying the product of powers rule:

$w^5 imes w^{-2} = w^{5 + (-2)} = w^{5-2} = w^3$

Thus, the $w$ part of the expression simplifies to $w^3$.

Step 6: Write the Simplified Expression

Now that we have simplified each part, we can combine them to write the fully simplified expression. We have:

• Coefficients: 36
• $x$ terms: 1
• $y$ terms: $y^8$
• $w$ terms: $w^3$

Multiplying these together gives us:

$36 imes 1 imes y^8 imes w^3 = 36y^8w^3$

This is the simplified expression with all positive exponents.

Detailed Explanation of Each Rule Applied

To fully grasp the simplification process, it's essential to understand how each rule of exponents is applied in detail. We've used the product of powers rule extensively, along with the negative exponent rule and the zero exponent rule. Let's delve deeper into these applications.

Product of Powers Rule: $a^m imes a^n = a^{m+n}$

The product of powers rule is a cornerstone in simplifying expressions with exponents. It states that when you multiply two powers with the same base, you add the exponents. This rule is derived from the fundamental definition of exponents, where $a^m$ means $a$ multiplied by itself $m$ times. So, when you multiply $a^m$ by $a^n$, you are essentially multiplying $a$ by itself $m+n$ times.

In our simplification, we applied this rule to the $x$, $y$, and $w$ terms:

• For $x$ terms: $x^{-1} imes x = x^{-1} imes x^1 = x^{-1+1} = x^0$
• For $y$ terms: $y^2 imes y^6 = y^{2+6} = y^8$
• For $w$ terms: $w^5 imes w^{-2} = w^{5 + (-2)} = w^3$

Each application involves identifying the common base (either $x$, $y$, or $w$) and then adding the exponents. This rule streamlines the multiplication of terms with the same base, making the simplification process more efficient.

Negative Exponent Rule: $a^{-n} = 1/a^n$

The negative exponent rule allows us to rewrite terms with negative exponents as fractions with positive exponents. This rule is crucial for our goal of expressing the final answer with only positive exponents. A negative exponent indicates the reciprocal of the base raised to the positive exponent.

In our expression, we encountered $x^{-1}$ and $w^{-2}$. The negative exponent rule transforms these terms:

• $x^{-1}$ can be thought of as $1/x^1$, which simplifies to $1/x$.
• $w^{-2}$ can be thought of as $1/w^2$.

However, in the context of the entire expression, we didn't need to explicitly rewrite $x^{-1}$ as $1/x$ because it canceled out with the $x^1$ term, resulting in $x^0 = 1$. Similarly, $w^{-2}$ was combined with $w^5$ using the product of powers rule, which resulted in a positive exponent: $w^3$.

Zero Exponent Rule: $a^0 = 1$

The zero exponent rule states that any non-zero number raised to the power of zero is 1. This rule is a special case derived from the quotient of powers rule. When you divide $a^n$ by itself ($a^n$), the result is 1. According to the quotient of powers rule, $a^n / a^n = a^{n-n} = a^0$. Therefore, $a^0$ must equal 1.

In our simplification, we used this rule when we combined the $x$ terms:

$x^{-1} imes x = x^{-1} imes x^1 = x^{-1+1} = x^0 = 1$

This application simplifies the expression by eliminating the $x$ term, as any term multiplied by 1 remains unchanged.

Common Mistakes to Avoid

When simplifying expressions with exponents, several common mistakes can occur. Recognizing and avoiding these pitfalls is essential for accurate simplification. Here are some mistakes to watch out for:

1. Incorrectly Applying the Product of Powers Rule: A common mistake is adding exponents when the bases are not the same. For example, $x^2 imes y^3$ cannot be simplified to $xy^5$ because the bases ($x$ and $y$) are different. The product of powers rule only applies when the bases are the same.
2. Misunderstanding Negative Exponents: Another frequent error is treating a term with a negative exponent as a negative number. For example, $x^{-2}$ is not equal to $-x^2$. Instead, $x^{-2}$ equals $1/x^2$. Always remember that a negative exponent indicates the reciprocal of the base raised to the positive exponent.
3. Forgetting the Zero Exponent Rule: Many students overlook the fact that any non-zero number raised to the power of zero is 1. For example, $5^0$ equals 1, not 0. Failing to apply this rule can lead to incorrect simplifications, especially when terms cancel each other out.
4. Incorrectly Distributing Exponents: When raising a product to a power, the exponent must be distributed to each factor. For example, $(xy)^2$ is $x^2y^2$, not $xy^2$. Similarly, when simplifying expressions with multiple terms inside parentheses, ensure that the exponent applies to all terms.
5. Arithmetic Errors: Simple arithmetic mistakes, such as adding or multiplying coefficients incorrectly, can lead to significant errors in the final result. Double-check your calculations, especially when dealing with multiple steps and terms.

By being mindful of these common mistakes and practicing regularly, you can improve your accuracy and confidence in simplifying expressions with exponents.

Practice Problems

To reinforce your understanding of simplifying expressions with exponents, working through practice problems is invaluable. Here are a few additional problems to try, along with brief solutions to help you check your work:

1. Simplify: $4a^3b^{-2} imes 2a^{-1}b^5$
   • Solution: $8a^2b^3$
2. Simplify: $(3x^2y^3)^2 imes (2xy^{-1})$
   • Solution: $18x^5y^5$
3. Simplify: rac{15p^4q^{-2}}{3p^{-1}q^3}
   • Solution: $5p^5q^{-5}$ or $5p^5/q^5$
4. Simplify: $5m^{-3}n^2 imes (2m^2n^{-1})^3$
   • Solution: $40m^3n^{-1}$ or $40m^3/n$

Working through these problems will solidify your understanding of the rules of exponents and help you develop a systematic approach to simplification. Remember to break down each problem into steps, apply the relevant rules, and double-check your work.

Conclusion

Simplifying expressions with positive exponents is a crucial skill in algebra and higher mathematics. By systematically applying the rules of exponents and avoiding common mistakes, you can efficiently manipulate complex expressions into their simplest forms. In this article, we walked through the simplification of $2 x^{-1} imes 3 y^2 imes 6 w^5 w^{-2} x y^6$, demonstrating each step in detail. Remember to group like terms, apply the product of powers rule, handle negative exponents carefully, and double-check your work. With practice, you'll become adept at simplifying expressions and building a solid foundation for more advanced mathematical concepts. The final simplified expression, $36y^8w^3$, showcases the power of these rules in action, providing a clear and concise representation of the original expression.