Simplifying Exponential Expressions A Step By Step Guide

Understanding exponential expressions is a fundamental concept in mathematics, particularly in algebra. These expressions involve a base raised to a power, and mastering the rules of exponents is crucial for simplifying and solving complex equations. In this article, we will delve into simplifying an expression involving exponents, providing a step-by-step guide and ensuring clarity for students and enthusiasts alike. We will address the question of which expression is equivalent to the given expression: $(3m^{-4})^3(3m^5)$. The solution requires a solid understanding of the power of a product rule, the power of a power rule, and the product of powers rule. Let's embark on this mathematical journey together and unravel the intricacies of exponential expressions.

Breaking Down the Expression

To simplify the expression $(3m^{-4})^3(3m^5)$, we need to apply the rules of exponents systematically. The first part of the expression, $(3m^{-4})^3$, involves raising a product to a power. According to the power of a product rule, $(ab)^n = a^n b^n$, we must raise each factor within the parentheses to the power of 3. This means we raise both 3 and $m^{-4}$ to the power of 3. Remember, the power of a product rule is a cornerstone in simplifying exponential expressions, and understanding its application is paramount. Next, we apply the power of a power rule, which states that $(a^m)^n = a^{mn}$. This rule dictates that when raising a power to another power, we multiply the exponents. So, $(m^{-4})^3$ becomes $m^{-4 imes 3} = m^{-12}$. Therefore, $(3m^{-4})^3$ simplifies to $3^3 m^{-12}$, which is $27m^{-12}$. This step is crucial as it sets the stage for further simplification by combining like terms. The methodical application of these rules ensures accuracy and helps in breaking down complex expressions into manageable components.

Applying the Rules of Exponents

Continuing with our simplification, we now have $27m^{-12}(3m^5)$. The next step involves applying the product of powers rule, which states that when multiplying exponential expressions with the same base, we add the exponents: $a^m imes a^n = a^{m+n}$. In our expression, we have $m^{-12}$ multiplied by $m^5$. Applying the product of powers rule, we add the exponents: $-12 + 5 = -7$. Therefore, $m^{-12} imes m^5$ simplifies to $m^{-7}$. Now, we must also multiply the coefficients, which are 27 and 3. Multiplying these gives us $27 imes 3 = 81$. Combining these results, we have $81m^{-7}$. This step is pivotal in bringing together the numerical coefficients and the variable terms with their respective exponents. The careful execution of this rule ensures that the expression is simplified correctly and efficiently, leading us closer to the final answer.

Eliminating Negative Exponents

Our expression is now $81m^{-7}$, but it is customary to express exponential expressions with positive exponents. To eliminate the negative exponent, we use the rule that a^{-n} = rac{1}{a^n}. Applying this rule to our expression, $m^{-7}$ becomes rac{1}{m^7}. Therefore, $81m^{-7}$ can be rewritten as 81 imes rac{1}{m^7}, which simplifies to rac{81}{m^7}. This transformation is essential for expressing the answer in its simplest form, adhering to mathematical conventions. Converting negative exponents to positive exponents not only simplifies the expression but also makes it easier to interpret and use in further calculations. This final step completes the simplification process, providing a clear and concise answer to the original question.

Final Answer and Conclusion

After systematically applying the rules of exponents, we have simplified the expression $(3m^{-4})^3(3m^5)$ to rac{81}{m^7}. This corresponds to option D in the given choices. To summarize, we first used the power of a product rule to expand $(3m^{-4})^3$ to $27m^{-12}$. Then, we applied the product of powers rule to multiply $27m^{-12}$ by $3m^5$, resulting in $81m^{-7}$. Finally, we eliminated the negative exponent by using the rule a^{-n} = rac{1}{a^n}, which gave us the final simplified expression of rac{81}{m^7}. This process demonstrates the importance of understanding and applying the rules of exponents accurately. By mastering these rules, students can confidently tackle more complex algebraic expressions. This comprehensive step-by-step solution not only provides the correct answer but also reinforces the underlying principles of exponential simplification.

Understanding the rules of exponents is crucial for simplifying algebraic expressions and solving equations effectively. These rules provide a systematic way to manipulate expressions involving powers, making them more manageable and easier to work with. In this section, we will delve into each rule used in simplifying the expression $(3m^{-4})^3(3m^5)$, providing detailed explanations and examples to ensure clarity and comprehension. We will cover the power of a product rule, the power of a power rule, the product of powers rule, and the rule for eliminating negative exponents. Each rule plays a vital role in the simplification process, and a thorough understanding of these rules is essential for success in algebra and beyond. By breaking down these concepts, we aim to equip learners with the tools they need to confidently approach and solve problems involving exponents.

Power of a Product Rule

The power of a product rule, expressed as $(ab)^n = a^n b^n$, is a fundamental concept in simplifying expressions. This rule states that when a product of two or more factors is raised to a power, each factor must be raised to that power individually. This is crucial because it allows us to distribute the exponent across the terms within the parentheses. In the context of our problem, we had the expression $(3m^{-4})^3$. Applying the power of a product rule, we raise both 3 and $m^{-4}$ to the power of 3. This gives us $3^3$ and $(m^{-4})^3$. The number 3 raised to the power of 3 is simply $3 imes 3 imes 3 = 27$. The term $(m^{-4})^3$ requires the application of another rule, the power of a power rule, which we will discuss next. Understanding the power of a product rule is not just about memorizing the formula; it’s about grasping the underlying principle that exponents distribute over multiplication. This foundational knowledge enables us to tackle more complex expressions with confidence and accuracy. Proper application of this rule ensures that each term within the parentheses is correctly accounted for when raised to a power.

Power of a Power Rule

The power of a power rule, written as $(a^m)^n = a^{mn}$, is another essential tool in simplifying exponential expressions. This rule states that when a power is raised to another power, we multiply the exponents. This rule is particularly useful when dealing with nested exponents, where one exponent is applied to a term that already has an exponent. In our expression, after applying the power of a product rule, we had the term $(m^{-4})^3$. Applying the power of a power rule, we multiply the exponents -4 and 3, which gives us $-4 imes 3 = -12$. Therefore, $(m^{-4})^3$ simplifies to $m^{-12}$. This step is crucial because it combines the exponents into a single term, making the expression easier to manage. It is important to note that the base remains the same while the exponents are multiplied. This rule is frequently used in conjunction with other exponent rules to simplify complex expressions. Mastering the power of a power rule allows for efficient simplification and is a key skill in algebraic manipulation. The correct application of this rule ensures that the exponents are handled accurately, leading to the correct simplified form.

Product of Powers Rule

The product of powers rule, expressed as $a^m imes a^n = a^{m+n}$, is vital for simplifying expressions involving the multiplication of terms with the same base. This rule states that when multiplying exponential expressions with the same base, we add the exponents. This rule is based on the fundamental principle that exponents represent repeated multiplication, and adding exponents is equivalent to combining these repeated multiplications. In our problem, after simplifying $(3m^{-4})^3$ to $27m^{-12}$, we needed to multiply this by $3m^5$. Applying the product of powers rule, we focus on the terms with the same base, which are $m^{-12}$ and $m^5$. Adding the exponents, we get $-12 + 5 = -7$. Therefore, $m^{-12} imes m^5$ simplifies to $m^{-7}$. Additionally, we multiply the coefficients, which are 27 and 3, to get 81. Combining these results, we have $81m^{-7}$. The product of powers rule is a cornerstone in simplifying exponential expressions, and its correct application is essential for achieving the simplest form. This rule simplifies the process of multiplying exponential terms by providing a direct method for combining exponents of like bases.

Eliminating Negative Exponents Rule

The rule for eliminating negative exponents, written as a^{-n} = rac{1}{a^n}, is essential for expressing exponential expressions in their simplest and most conventional form. A negative exponent indicates the reciprocal of the base raised to the positive exponent. This rule allows us to rewrite terms with negative exponents as fractions with positive exponents, which is generally preferred in mathematical notation. In our simplification, we arrived at the expression $81m^{-7}$. To eliminate the negative exponent, we apply the rule, rewriting $m^{-7}$ as rac{1}{m^7}. Therefore, $81m^{-7}$ becomes 81 imes rac{1}{m^7}, which simplifies to rac{81}{m^7}. This final step ensures that the expression is presented in a standard format, making it easier to interpret and use in further calculations. The elimination of negative exponents is a crucial step in the simplification process, ensuring that the final answer is expressed in its most readable and practical form. Understanding and applying this rule correctly is a fundamental aspect of working with exponential expressions.

A step-by-step solution walkthrough is invaluable for grasping the intricacies of simplifying complex algebraic expressions. By breaking down the process into manageable steps, we can ensure clarity and understanding at each stage. In this section, we will revisit the original problem, $(3m^{-4})^3(3m^5)$, and provide a detailed, step-by-step solution. Each step will be explained thoroughly, referencing the relevant exponent rules used and the reasoning behind each manipulation. This approach is designed to reinforce the concepts discussed earlier and provide a practical guide for solving similar problems. We will cover each rule application in sequence, from the power of a product rule to the elimination of negative exponents. By following this walkthrough, learners can gain confidence in their ability to simplify exponential expressions and tackle more challenging problems. This detailed breakdown is a crucial tool for both students and educators in mastering the art of simplifying algebraic expressions.

Step 1: Applying the Power of a Product Rule

The first step in simplifying the expression $(3m^{-4})^3(3m^5)$ involves applying the power of a product rule. This rule states that $(ab)^n = a^n b^n$, meaning we must raise each factor within the parentheses to the power of 3. Therefore, $(3m^{-4})^3$ becomes $3^3 (m^{-4})^3$. This initial step is critical as it sets the foundation for further simplification. By correctly distributing the exponent across the terms within the parentheses, we ensure that each term is properly accounted for in the subsequent steps. Ignoring this step or applying it incorrectly can lead to significant errors in the final result. It is important to remember that this rule applies to any number of factors within the parentheses, not just two. The methodical application of the power of a product rule is a key skill in simplifying complex expressions and is essential for achieving accuracy.

Step 2: Applying the Power of a Power Rule

Following the application of the power of a product rule, our expression is now $3^3 (m^{-4})^3 (3m^5)$. The next step involves applying the power of a power rule, which states that $(a^m)^n = a^{mn}$. We focus on the term $(m^{-4})^3$, where we need to multiply the exponents. Multiplying -4 by 3 gives us -12. So, $(m^{-4})^3$ simplifies to $m^{-12}$. Now our expression becomes $3^3 m^{-12} (3m^5)$. Evaluating $3^3$ gives us 27, so we have $27m^{-12}(3m^5)$. This step is crucial as it reduces the complexity of the expression by combining exponents. Misapplication of this rule can lead to incorrect exponent values and a flawed final answer. The power of a power rule is a fundamental concept in exponent manipulation and must be applied carefully to ensure accuracy.

Step 3: Applying the Product of Powers Rule

After simplifying the powers, our expression is $27m^{-12}(3m^5)$. The next step involves applying the product of powers rule, which states that $a^m imes a^n = a^{m+n}$. We multiply the coefficients 27 and 3, which gives us 81. Then, we multiply the terms with the same base, $m^{-12}$ and $m^5$. Adding the exponents, we get $-12 + 5 = -7$. Therefore, $m^{-12} imes m^5$ simplifies to $m^{-7}$. Combining these results, our expression becomes $81m^{-7}$. This step is essential for consolidating like terms and simplifying the expression further. The product of powers rule is a core concept in exponent manipulation, and its correct application is vital for achieving the simplest form of the expression. It is important to ensure that only terms with the same base have their exponents added.

Step 4: Eliminating the Negative Exponent

Our expression is now $81m^{-7}$. The final step involves eliminating the negative exponent using the rule a^{-n} = rac{1}{a^n}. We rewrite $m^{-7}$ as rac{1}{m^7}. Therefore, $81m^{-7}$ becomes 81 imes rac{1}{m^7}, which simplifies to rac{81}{m^7}. This is the final simplified form of the expression. Eliminating negative exponents is crucial for expressing the answer in its most conventional and easily understandable form. This step ensures that the expression is presented in a standard mathematical format, making it easier to interpret and use in subsequent calculations. The correct application of this rule completes the simplification process, providing a clear and concise final answer.

Conclusion of the Step-by-Step Solution

In conclusion, by systematically applying the rules of exponents, we have successfully simplified the expression $(3m^{-4})^3(3m^5)$ to rac{81}{m^7}. We began by applying the power of a product rule, followed by the power of a power rule, then the product of powers rule, and finally, we eliminated the negative exponent. Each step was crucial in transforming the original expression into its simplest form. This step-by-step walkthrough highlights the importance of understanding and correctly applying the rules of exponents. By mastering these rules and following a systematic approach, students can confidently simplify complex algebraic expressions. This detailed solution not only provides the correct answer but also reinforces the underlying principles of exponential simplification, empowering learners to tackle similar problems with ease and accuracy.

Avoiding common mistakes is a critical aspect of mastering any mathematical concept, and simplifying exponential expressions is no exception. There are several pitfalls that students often encounter when working with exponents, which can lead to incorrect answers and a misunderstanding of the underlying principles. In this section, we will discuss some of the most common mistakes made when simplifying expressions like $(3m^{-4})^3(3m^5)$, providing clear explanations and strategies for avoiding them. These mistakes often involve misapplication of exponent rules, incorrect handling of negative exponents, and errors in arithmetic calculations. By understanding these common errors, students can develop a more robust understanding of exponent rules and improve their problem-solving skills. This discussion aims to equip learners with the knowledge and awareness necessary to confidently and accurately simplify exponential expressions.

Misapplying the Power of a Product Rule

One of the most common mistakes in simplifying exponential expressions is misapplying the power of a product rule. This rule, $(ab)^n = a^n b^n$, requires that the exponent be applied to each factor within the parentheses. A frequent error is applying the exponent to only one factor or incorrectly distributing it. For example, in the expression $(3m^{-4})^3$, some students might incorrectly calculate it as $3(m^{-4})^3$ or $3^3m^{-4}$ instead of the correct $3^3 (m^{-4})^3$. This mistake stems from a misunderstanding of how the exponent distributes across multiplication. To avoid this error, it is crucial to remember that the exponent affects every factor within the parentheses, whether it is a numerical coefficient or a variable term. Double-checking that each factor has been raised to the correct power is a simple yet effective way to prevent this mistake. Correct application of the power of a product rule is fundamental for accurate simplification of exponential expressions.

Incorrectly Applying the Power of a Power Rule

Another frequent mistake is incorrectly applying the power of a power rule, which states that $(a^m)^n = a^{mn}$. The common error here is adding the exponents instead of multiplying them. For instance, in the term $(m^{-4})^3$, some students might incorrectly calculate it as $m^{-4+3} = m^{-1}$ instead of the correct $m^{-4 imes 3} = m^{-12}$. This mistake arises from confusing the power of a power rule with the product of powers rule, where exponents are added. To avoid this error, it is essential to remember that when a power is raised to another power, the exponents are multiplied, not added. Reinforcing this distinction through practice and careful attention to the rule will help prevent this common mistake. Accurate application of the power of a power rule is crucial for simplifying exponential expressions correctly.

Mixing Up the Product of Powers Rule

Mixing up the product of powers rule is another common source of errors in simplifying exponential expressions. This rule, $a^m imes a^n = a^{m+n}$, is often confused with other exponent rules, particularly the power of a power rule. A typical mistake is multiplying the exponents instead of adding them when multiplying terms with the same base. For example, students might incorrectly simplify $m^{-12} imes m^5$ as $m^{-12 imes 5} = m^{-60}$ instead of the correct $m^{-12+5} = m^{-7}$. This confusion often stems from a lack of clear understanding of the conditions under which each rule applies. To prevent this error, it is vital to remember that the product of powers rule applies specifically when multiplying exponential terms with the same base, and the exponents should be added in this case. Regular practice and a clear understanding of the rule's conditions will help minimize this mistake.

Mishandling Negative Exponents

Mishandling negative exponents is a pervasive issue in simplifying exponential expressions. The rule a^{-n} = rac{1}{a^n} is often misunderstood or misapplied, leading to incorrect simplifications. A common mistake is to treat the negative exponent as a negative sign applied to the base, rather than as an indication of a reciprocal. For example, some students might incorrectly rewrite $m^{-7}$ as $-m^7$ instead of the correct rac{1}{m^7}. This misunderstanding can lead to significant errors in the final answer. To avoid this mistake, it is crucial to remember that a negative exponent indicates the reciprocal of the base raised to the positive exponent. Regular practice with negative exponents and a clear understanding of the rule will help prevent this error. Accurate handling of negative exponents is essential for correct simplification of exponential expressions.

Arithmetic Errors in Calculations

Finally, arithmetic errors in calculations can also lead to mistakes in simplifying exponential expressions. These errors can occur when multiplying coefficients, adding or multiplying exponents, or performing other numerical operations. For example, in the expression $27m^{-12}(3m^5)$, an arithmetic error might occur when multiplying 27 and 3. Similarly, errors can happen when adding exponents, such as in the step $-12 + 5$. These errors, although seemingly minor, can propagate through the rest of the simplification process and result in an incorrect final answer. To minimize arithmetic errors, it is important to double-check each calculation and use a systematic approach. Breaking down complex calculations into smaller steps can also help reduce the likelihood of mistakes. While a strong understanding of exponent rules is essential, careful attention to arithmetic details is equally important for accurate simplification.

Practice problems and solutions are an indispensable tool for mastering any mathematical concept, and simplifying exponential expressions is no different. By working through a variety of problems, students can reinforce their understanding of the rules and develop the problem-solving skills necessary to tackle more complex expressions. In this section, we will provide several practice problems covering the various exponent rules discussed earlier, along with detailed solutions. These problems are designed to challenge learners and help them solidify their understanding of the concepts. Each problem will be followed by a step-by-step solution, providing clear explanations of the methods used and the reasoning behind each step. This practical application of the rules is crucial for building confidence and competence in simplifying exponential expressions. The problems will range in difficulty, allowing students to progressively build their skills and understanding.

Practice Problem 1

Simplify the expression: $(2x^3y^{-2})^4$

Solution

1. Apply the power of a product rule: $(2x^3y^{-2})^4 = 2^4 (x^3)^4 (y^{-2})^4$
2. Simplify $2^4$: $2^4 = 16$
3. Apply the power of a power rule: $(x^3)^4 = x^{3 imes 4} = x^{12}$ and $(y^{-2})^4 = y^{-2 imes 4} = y^{-8}$
4. Rewrite the expression: $16x^{12}y^{-8}$
5. Eliminate the negative exponent: y^{-8} = rac{1}{y^8}
6. Final simplified expression: rac{16x^{12}}{y^8}

This problem tests the understanding of both the power of a product rule and the power of a power rule. By applying these rules systematically, the expression can be simplified step-by-step.

Practice Problem 2

Simplify the expression: rac{a^5b^{-3}}{a^2b^2}

Solution

1. Apply the quotient of powers rule: rac{a^5}{a^2} = a^{5-2} = a^3 and rac{b^{-3}}{b^2} = b^{-3-2} = b^{-5}
2. Rewrite the expression: $a^3b^{-5}$
3. Eliminate the negative exponent: b^{-5} = rac{1}{b^5}
4. Final simplified expression: rac{a^3}{b^5}

This problem focuses on the application of the quotient of powers rule and the elimination of negative exponents. Understanding how to handle division with exponents is crucial here.

Practice Problem 3

Simplify the expression: $(4m^{-2}n^5)(3m^4n^{-1})$

Solution

1. Multiply the coefficients: $4 imes 3 = 12$
2. Apply the product of powers rule for m: $m^{-2} imes m^4 = m^{-2+4} = m^2$
3. Apply the product of powers rule for n: $n^5 imes n^{-1} = n^{5-1} = n^4$
4. Final simplified expression: $12m^2n^4$

This problem combines the product of powers rule with the multiplication of coefficients. It emphasizes the importance of handling each part of the expression systematically.

Practice Problem 4

Simplify the expression: rac{(x^2y^{-1})^3}{x^{-2}y^4}

Solution

1. Apply the power of a product rule in the numerator: $(x^2y^{-1})^3 = (x^2)^3 (y^{-1})^3 = x^6y^{-3}$
2. Rewrite the expression: rac{x^6y^{-3}}{x^{-2}y^4}
3. Apply the quotient of powers rule: rac{x^6}{x^{-2}} = x^{6-(-2)} = x^8 and rac{y^{-3}}{y^4} = y^{-3-4} = y^{-7}
4. Rewrite the expression: $x^8y^{-7}$
5. Eliminate the negative exponent: y^{-7} = rac{1}{y^7}
6. Final simplified expression: rac{x^8}{y^7}

This problem is a comprehensive exercise that combines the power of a product rule, the quotient of powers rule, and the elimination of negative exponents. It requires a thorough understanding of all the exponent rules.

Importance of Practice

These practice problems and solutions provide a valuable resource for mastering the simplification of exponential expressions. By working through these problems and reviewing the solutions, students can reinforce their understanding of the rules and develop the skills necessary to tackle more complex expressions. The key to mastering these concepts is consistent practice and a systematic approach to problem-solving. Each problem should be approached methodically, applying the relevant rules step-by-step to ensure accuracy. With regular practice, simplifying exponential expressions can become a straightforward and manageable task.