Simplifying Expressions Using Kelly's Steps: A Step-by-Step Guide

In the realm of mathematics, simplifying expressions is a fundamental skill. It allows us to take complex-looking equations and reduce them to their most basic, understandable forms. This not only makes them easier to work with but also reveals the underlying relationships between variables and constants. One particularly useful technique for simplifying expressions involving exponents is Kelly's steps, which provide a systematic approach to tackling these problems. In this article, we'll delve into Kelly's steps and apply them to simplify the expression $\left(5 w^7\right)^2$. Furthermore, we'll explore the concept of the power of a product and identify the correct simplified form from a set of options.

Understanding Kelly's Steps

Kelly's steps, often used in the context of simplifying expressions with exponents, involve a series of rules and properties that allow us to manipulate and reduce complex expressions. These steps are rooted in the fundamental laws of exponents, which govern how exponents interact with multiplication, division, and other operations. By understanding and applying these rules, we can systematically simplify expressions and arrive at their most concise forms.

The core principles underlying Kelly's steps include:

• Power of a product: This rule states that when a product is raised to a power, each factor within the product is raised to that power. Mathematically, this is expressed as $(ab)^n = a^n b^n$.
• Power of a power: This rule states that when a power is raised to another power, the exponents are multiplied. Mathematically, this is expressed as $(a^m)^n = a^{m \cdot n}$.
• Product of powers: This rule states that when multiplying powers with the same base, the exponents are added. Mathematically, this is expressed as $a^m \cdot a^n = a^{m+n}$.

By applying these rules in a strategic sequence, we can effectively simplify expressions involving exponents.

Applying Kelly's Steps to $\left(5 w^7\right)^2$

Let's now apply Kelly's steps to simplify the expression $\left(5 w^7\right)^2$. This expression involves a product, $5w^7$, raised to the power of 2. To simplify this, we'll use the power of a product rule, which states that $(ab)^n = a^n b^n$. Applying this rule, we get:

$\left(5 w^7\right)^2 = 5^2 \cdot (w^7)^2$

Next, we need to simplify $5^2$ and $(w^7)^2$. The first term, $5^2$, is simply 5 multiplied by itself, which equals 25. The second term, $(w^7)^2$, involves a power raised to another power. To simplify this, we'll use the power of a power rule, which states that $(a^m)^n = a^{m \cdot n}$. Applying this rule, we get:

$(w^7)^2 = w^{7 \cdot 2} = w^{14}$

Now, we can substitute these simplified terms back into our expression:

$5^2 \cdot (w^7)^2 = 25 \cdot w^{14}$

Therefore, the simplified form of $\left(5 w^7\right)^2$ is $25w^{14}$.

Identifying the Correct Simplified Form

Now, let's consider the question of identifying the correct simplified form from the given options:

• $5 w^9$
• $10 w^{14}$
• $25 w^9$
• $25 w^{14}$

Based on our simplification using Kelly's steps, we found that $\left(5 w^7\right)^2$ simplifies to $25w^{14}$. Therefore, the correct simplified form is $25 w^{14}$. The other options are incorrect because they either misapply the power of a product rule or the power of a power rule.

The Power of a Product: A Deeper Dive

The power of a product rule, as we've seen, is a crucial tool in simplifying expressions with exponents. It's worth exploring this rule in more detail to fully grasp its implications. The rule essentially states that when a product of two or more factors is raised to a power, each factor is raised to that power individually. This can be expressed mathematically as:

$(ab)^n = a^n b^n$

This rule is not limited to two factors; it can be extended to any number of factors within the product. For example, if we have $(abc)^n$, we can apply the power of a product rule to get:

$(abc)^n = a^n b^n c^n$

This rule is a direct consequence of the definition of exponents. When we raise a product to a power, we are essentially multiplying the product by itself a certain number of times. For example, $(ab)^3$ means $(ab)(ab)(ab)$. By rearranging the factors, we can write this as $(a \cdot a \cdot a)(b \cdot b \cdot b)$, which is equal to $a^3 b^3$.

Understanding the power of a product rule is essential for simplifying a wide range of expressions, including those involving polynomials, rational expressions, and more. It allows us to break down complex expressions into simpler components, making them easier to manipulate and solve.

Common Mistakes to Avoid

When simplifying expressions with exponents, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accurate simplification.

One common mistake is incorrectly applying the power of a product rule. For example, some students might mistakenly think that $(a+b)^n$ is equal to $a^n + b^n$. However, this is incorrect. The power of a product rule applies only to products, not sums or differences. To expand $(a+b)^n$, you would need to use the binomial theorem or multiply $(a+b)$ by itself $n$ times.

Another common mistake is confusing the power of a power rule with the product of powers rule. The power of a power rule states that $(a^m)^n = a^{m \cdot n}$, while the product of powers rule states that $a^m \cdot a^n = a^{m+n}$. It's crucial to remember that when raising a power to another power, you multiply the exponents, but when multiplying powers with the same base, you add the exponents.

A third common mistake is forgetting to apply the exponent to all factors within the product. For example, in the expression $(2x)^3$, some students might correctly raise $x$ to the power of 3, but forget to raise 2 to the power of 3 as well. The correct simplification is $(2x)^3 = 2^3 x^3 = 8x^3$.

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

Practice Problems

To solidify your understanding of simplifying expressions with Kelly's steps and the power of a product rule, let's work through a few practice problems.

Problem 1: Simplify the expression $\left(3x^2y^5\right)^3$.

Solution:

Using the power of a product rule, we get:

$\left(3x^2y^5\right)^3 = 3^3 (x^2)^3 (y^5)^3$

Simplifying each term, we have:

$3^3 = 27$

$(x^2)^3 = x^{2 \cdot 3} = x^6$

$(y^5)^3 = y^{5 \cdot 3} = y^{15}$

Therefore, the simplified expression is $27x^6y^{15}$.

Problem 2: Simplify the expression $\left(\frac{2a^4}{b^3}\right)^2$.

Solution:

Using the power of a product rule and the power of a quotient rule (which is a variation of the power of a product rule), we get:

$\left(\frac{2a^4}{b^3}\right)^2 = \frac{(2a^4)^2}{(b^3)^2}$

Simplifying the numerator and denominator separately, we have:

$(2a^4)^2 = 2^2 (a^4)^2 = 4a^8$

$(b^3)^2 = b^{3 \cdot 2} = b^6$

Therefore, the simplified expression is $\frac{4a^8}{b^6}$.

Problem 3: Simplify the expression $\left(-5m^3n\right)^4$.

Solution:

Using the power of a product rule, we get:

$\left(-5m^3n\right)^4 = (-5)^4 (m^3)^4 n^4$

Simplifying each term, we have:

$(-5)^4 = 625$

$(m^3)^4 = m^{3 \cdot 4} = m^{12}$

Therefore, the simplified expression is $625m^{12}n^4$.

By working through these practice problems, you can further develop your skills in simplifying expressions with exponents and the power of a product rule.

Conclusion

Simplifying expressions with exponents is a fundamental skill in mathematics. Kelly's steps, based on the laws of exponents, provide a systematic approach to tackling these problems. The power of a product rule, in particular, is a powerful tool for simplifying expressions where a product is raised to a power. By understanding and applying these concepts, you can confidently simplify complex expressions and solve a wide range of mathematical problems. Remember to practice regularly and be mindful of common mistakes to ensure accuracy and proficiency in simplifying expressions with exponents.

Simplifying Expressions with Kelly's Steps: Mastering Powers of Products

Are you struggling with simplifying expressions, especially those involving exponents? Kelly's steps provide a clear and effective method for tackling these problems. This comprehensive guide will walk you through Kelly's steps, focusing on the crucial concept of the power of a product and how to apply it to simplify complex expressions. We'll use the example of simplifying the expression $\left(5 w^7\right)^2$ to illustrate the process and then explore how to identify the correct simplified form from a set of options. By the end of this article, you'll have a solid understanding of how to use Kelly's steps to confidently simplify expressions involving exponents.

Unveiling Kelly's Steps: A Systematic Approach

At the heart of simplifying expressions with exponents lies a set of principles known as Kelly's steps. These steps are not a rigid formula but rather a systematic way of thinking about and applying the fundamental laws of exponents. The power of this approach lies in its ability to break down complex problems into manageable steps, leading to accurate and efficient solutions. Mastering Kelly's steps involves understanding the core exponent rules and strategically applying them to simplify expressions. Let's delve into the key principles that underpin Kelly's steps and how they form the foundation for simplifying complex algebraic expressions.

The essence of Kelly's steps rests on three fundamental exponent rules:

1. Power of a product rule: This rule is a cornerstone of simplification and states that when a product of factors is raised to a power, each factor is individually raised to that power. In mathematical notation, this is represented as $(ab)^n = a^n b^n$. This rule is essential for distributing the exponent across multiple terms within parentheses, as we'll see in our example. Understanding this rule thoroughly is the first step in mastering Kelly's approach. When you encounter an expression where a product is raised to a power, remember that the exponent applies to every factor within the parentheses. This will prevent common errors and ensure accurate simplification.

2. Power of a power rule: This rule addresses situations where an exponent is raised to another exponent. It states that when you have a power raised to another power, you multiply the exponents. This can be mathematically expressed as $(a^m)^n = a^{m \cdot n}$. This rule is critical for simplifying terms like $(w^7)^2$, where we need to multiply the exponents 7 and 2. The power of a power rule is a powerful tool for collapsing nested exponents into a single, simplified term. Keep in mind that this rule only applies when you have an exponent raised to another exponent, not when you are multiplying terms with the same base.

3. Product of powers rule: While not directly used in our example, this rule is equally important in the broader context of simplifying expressions. It states that when multiplying powers with the same base, you add the exponents. This is expressed as $a^m \cdot a^n = a^{m+n}$. Imagine you are multiplying $x^3$ by $x^2$, according to this rule you would add the exponent which results in $x^5$. Remember, it's crucial to add exponents when multiplying powers that share the same base. This rule often comes into play when simplifying expressions after applying the power of a product or power of a power rule. The product of powers rule is a valuable tool in your algebraic toolbox.

These three rules, when applied strategically, form the backbone of Kelly's steps for simplifying expressions with exponents. By mastering these rules, you'll be well-equipped to tackle a wide range of simplification problems. Let's now apply these principles to our specific example, demonstrating how Kelly's steps can be used in practice.

Applying Kelly's Steps: Simplifying $\left(5 w^7\right)^2$ Step-by-Step

Now, let's put Kelly's steps into action by simplifying the expression $\left(5 w^7\right)^2$. This expression represents a product, $5w^7$, raised to the power of 2. To unravel this, we will strategically apply the exponent rules we discussed earlier. This step-by-step simplification will not only provide the correct answer but also illustrate the methodical approach that Kelly's steps encourage. Follow along carefully as we break down the process, highlighting the key decision points and the application of the relevant exponent rules.

The first crucial step is to recognize that we have a product, $5w^7$, enclosed in parentheses and raised to the power of 2. This immediately signals the applicability of the power of a product rule. Remember, this rule allows us to distribute the exponent to each factor within the product. Applying this rule to our expression, we obtain:

$\left(5 w^7\right)^2 = 5^2 \cdot (w^7)^2$

Notice how the exponent 2 has been applied to both the constant factor, 5, and the variable term, $w^7$. This is the essence of the power of a product rule – every factor within the parentheses gets the exponent. This step is crucial for setting up the rest of the simplification process. Now, we have two simpler terms to deal with: $5^2$ and $(w^7)^2$. The key is to handle each term individually, using the appropriate exponent rule.

Next, we focus on simplifying the individual terms. The first term, $5^2$, is straightforward. It simply means 5 multiplied by itself, which equals 25. So, we have:

$5^2 = 5 \cdot 5 = 25$

The second term, $(w^7)^2$, presents an exponent raised to another exponent. This is a clear indication that we should apply the power of a power rule. Recall that this rule states that when you have a power raised to another power, you multiply the exponents. Applying this rule to $(w^7)^2$, we get:

$(w^7)^2 = w^{7 \cdot 2} = w^{14}$

Now, we have successfully simplified both individual terms. The final step is to combine these simplified terms back into a single expression. Substituting our simplified results back into the equation, we get:

$25 \cdot w^{14} = 25w^{14}$

Therefore, the simplified form of $\left(5 w^7\right)^2$ is $25w^{14}$. This step-by-step approach demonstrates the power of Kelly's steps in breaking down a seemingly complex problem into a series of manageable operations. By carefully applying the exponent rules, we have arrived at the simplified expression.

Identifying the Correct Simplified Form: A Multiple-Choice Challenge

Let's reinforce our understanding by examining how to identify the correct simplified form from a set of options. This is a common type of problem you might encounter in mathematics assessments. Given our simplified expression, $25w^{14}$, we'll analyze a set of multiple-choice answers and pinpoint the correct one. This exercise will solidify your ability to not only simplify expressions but also to recognize the correct result in different formats.

Consider the following options:

• $5 w^9$
• $10 w^{14}$
• $25 w^9$
• $25 w^{14}$

We know from our previous simplification that the correct answer is $25w^{14}$. Let's analyze why the other options are incorrect:

• $5 w^9$: This option incorrectly applies the power of a product rule. It seems to have only squared the $w^7$ term but not the constant 5. Additionally, the exponent multiplication is also incorrect; 7 multiplied by 2 is 14, not 9. Spotting this type of error is crucial for avoiding mistakes.
• $10 w^{14}$: This option also demonstrates a misunderstanding of the power of a product rule. It appears that the constant 5 was multiplied by 2 instead of being squared. This highlights the importance of remembering that exponents indicate repeated multiplication, not addition. Remember the fundamental operations associated with exponents.
• $25 w^9$: This option correctly squares the constant 5 but makes the same mistake as the first option with the exponent of w. It squares the base $5$ but incorrectly calculated the exponent. This shows that applying the rules partially is not sufficient; we must apply all rules to get the correct result.
• $25 w^{14}$: This option perfectly matches our simplified expression. It correctly applies both the power of a product rule and the power of a power rule, resulting in the accurate simplification.

Therefore, the correct simplified form is, undoubtedly, $25 w^{14}$. This analysis emphasizes the importance of careful application of Kelly's steps and a thorough understanding of the exponent rules. By systematically working through the problem and double-checking each step, we can confidently identify the correct answer.

Decoding the Power of a Product Rule: A Fundamental Principle

The power of a product rule is a cornerstone of simplifying expressions with exponents. It's so crucial that it's worth exploring in more detail to solidify our understanding. This rule, which we've used extensively in our example, provides a powerful way to distribute exponents across multiple factors within a product. Understanding the underlying principle behind this rule will empower you to apply it confidently in various situations. Let's delve deeper into the mathematical foundation and practical applications of the power of a product rule.

The power of a product rule, stated simply, allows us to distribute an exponent over a product of factors. This means that if we have an expression like $(ab)^n$, where a and b are any algebraic expressions and n is an exponent, we can rewrite it as $a^n b^n$. This distribution is the key to simplifying expressions where multiple factors are raised to a power. This rule can be extended to any number of factors within the product. For instance, $(abc)^n$ would become $a^n b^n c^n$. This broad applicability makes the power of a product rule an indispensable tool in algebra.

To grasp the essence of this rule, it's helpful to think about what an exponent really means. An exponent indicates repeated multiplication. So, $(ab)^n$ means multiplying the product (ab) by itself n times: $(ab)(ab)(ab)...$ (n times). Now, due to the commutative and associative properties of multiplication, we can rearrange the order of the factors: $(a \cdot a \cdot a...)(b \cdot b \cdot b...)$, which is precisely $a^n b^n$. This intuitive understanding reinforces why the power of a product rule works.

Let's consider some examples to further illustrate the power of a product rule:

• (Example 1:) Simplify $(2xy^2)^3$. Applying the power of a product rule, we distribute the exponent 3 to each factor: $2^3 \cdot x^3 \cdot (y^2)^3$. This simplifies to $8x^3y^6$ after applying the power of a power rule.
• (Example 2:) Simplify $(-3a^2b)^2$. Distributing the exponent, we get $(-3)^2 \cdot (a^2)^2 \cdot b^2$. This simplifies to $9a^4b^2$ after evaluating $(-3)^2$ and applying the power of a power rule.

These examples showcase how the power of a product rule streamlines the simplification process. By distributing the exponent, we break down the complex expression into manageable terms. Mastering this rule is not just about memorizing a formula; it's about understanding the fundamental principle of distributing exponents over products.

Avoiding Common Pitfalls: Mistakes to Watch Out For

While Kelly's steps and the power of a product rule offer a powerful framework for simplifying expressions, it's crucial to be aware of common mistakes. Avoiding these pitfalls will significantly improve your accuracy and prevent frustration. Understanding common errors is as important as knowing the rules themselves. Let's explore some frequent mistakes and how to sidestep them.

One of the most common mistakes is misapplying the power of a product rule to sums or differences. Remember this crucial distinction: the power of a product rule applies ONLY to products, NOT sums or differences. For example, $(a+b)^n$ is NOT equal to $a^n + b^n$. This is a frequent error, so be extra cautious when dealing with expressions involving addition or subtraction within parentheses. To expand $(a+b)^n$, you would need to use the binomial theorem or multiply $(a+b)$ by itself n times, not directly distribute the exponent. Always double-check that the operation within the parentheses is multiplication before applying the power of a product rule.

Another frequent mistake is confusing the power of a power rule with the product of powers rule. These rules sound similar, but they have distinct applications. As we discussed earlier, the power of a power rule states that $(a^m)^n = a^{m \cdot n}$, meaning you multiply the exponents. The product of powers rule, on the other hand, states that $a^m \cdot a^n = a^{m+n}$, meaning you add the exponents when multiplying powers with the same base. The key difference is the operation. When raising a power to another power, multiply. When multiplying powers with the same base, add. Mixing up these rules will lead to incorrect simplifications, so take extra care to apply the correct rule based on the context of the expression.

A third common mistake is overlooking the exponent on a coefficient within the product. When distributing an exponent using the power of a product rule, it's essential to apply the exponent to every factor, including numerical coefficients. For instance, in the expression $(2x)^3$, students sometimes correctly raise x to the power of 3 but forget to cube the 2. The correct simplification is $(2x)^3 = 2^3 \cdot x^3 = 8x^3$. Don't let the constant terms slip your mind; they must be raised to the exponent as well. Developing the habit of systematically distributing the exponent to each factor, one by one, will help prevent this oversight.

By being mindful of these common pitfalls, you can significantly enhance your accuracy in simplifying expressions with exponents. Attention to detail and a thorough understanding of the exponent rules are your best defenses against these errors.

Practice Makes Perfect: Sharpen Your Skills with Examples

The best way to solidify your understanding of Kelly's steps and the power of a product rule is through practice. Working through examples will not only reinforce the concepts but also help you develop problem-solving strategies. Consistent practice is key to mastering any mathematical skill. Let's tackle a few practice problems together, walking through the solutions step-by-step to solidify your knowledge and build confidence.

Problem 1: Simplify the expression $\left(4a^3b^2\right)^2$.

Solution:

1. Apply the power of a product rule: Distribute the exponent 2 to each factor within the parentheses: $4^2 \cdot (a^3)^2 \cdot (b^2)^2$.
2. Simplify each term: $4^2 = 16$. $(a^3)^2 = a^{3 \cdot 2} = a^6$ (using the power of a power rule). $(b^2)^2 = b^{2 \cdot 2} = b^4$ (using the power of a power rule).
3. Combine the simplified terms: $16a^6b^4$.

Therefore, the simplified expression is $16a^6b^4$.

Problem 2: Simplify the expression $\left(\frac{3x^2}{y^4}\right)^3$.

Solution:

1. Apply the power of a product rule (and the power of a quotient rule, which is a variation of the power of a product rule): Distribute the exponent 3 to both the numerator and the denominator: $\frac{(3x^2)^3}{(y^4)^3}$.
2. Simplify the numerator: $(3x^2)^3 = 3^3 \cdot (x^2)^3 = 27x^6$ (using the power of a product and power of a power rules).
3. Simplify the denominator: $(y^4)^3 = y^{4 \cdot 3} = y^{12}$ (using the power of a power rule).
4. Combine the simplified terms: $\frac{27x^6}{y^{12}}$.

Therefore, the simplified expression is $\frac{27x^6}{y^{12}}$.

Problem 3: Simplify the expression $\left(-2m^4n^3\right)^5$.

Solution:

1. Apply the power of a product rule: Distribute the exponent 5 to each factor within the parentheses: $(-2)^5 \cdot (m^4)^5 \cdot (n^3)^5$.
2. Simplify each term: $(-2)^5 = -32$. $(m^4)^5 = m^{4 \cdot 5} = m^{20}$ (using the power of a power rule). $(n^3)^5 = n^{3 \cdot 5} = n^{15}$ (using the power of a power rule).
3. Combine the simplified terms: $-32m^{20}n^{15}$.

Therefore, the simplified expression is $-32m^{20}n^{15}$.

These examples illustrate the consistent application of Kelly's steps and the power of a product rule. By breaking down the problems into manageable steps, you can confidently simplify even complex expressions. Continue practicing with different examples to refine your skills and build your mathematical fluency.

Conclusion: Mastering Simplification with Kelly's Steps

Simplifying expressions with exponents is a fundamental skill in algebra, and Kelly's steps provide a robust framework for mastering this skill. By understanding and applying the exponent rules, particularly the power of a product rule, you can confidently tackle a wide range of simplification problems. This comprehensive guide has walked you through the key concepts, practical applications, common pitfalls, and practice examples needed to excel in this area. Remember, the key to success lies in consistent practice and a methodical approach. By embracing Kelly's steps as a strategic tool, you can unlock your full potential in simplifying expressions and excel in your mathematical journey.

With a solid grasp of Kelly's steps and the power of a product rule, you are well-equipped to simplify complex expressions involving exponents. Embrace the challenges, continue practicing, and watch your algebraic skills soar. The journey to mathematical mastery is paved with consistent effort and a dedication to understanding fundamental principles. Keep practicing, keep learning, and keep simplifying!

Rewrite the Keywords

Original keywords: Simplify the expression $\left(5 w^7\right)^2$, simplified power of the product, $5 w^9$, $10 w^{14}$, $25 w^9$, $25 w^{14}$.

Rewritten keywords: How do you simplify the expression $\left(5 w^7\right)^2$ using Kelly's steps? What is the simplified form of the expression after applying the power of a product rule? Which of the following is the correct simplification of $\left(5 w^7\right)^2$: $5 w^9$, $10 w^{14}$, $25 w^9$, or $25 w^{14}$? What are Kelly's steps for simplifying expressions with exponents? What is the power of a product rule and how is it applied in simplifying algebraic expressions?