Rewrite (5⁻⁸)(5⁻¹⁰) As 5ⁿ A Step-by-Step Guide
In this comprehensive exploration, we will delve into the realm of exponential expressions and embark on a journey to rewrite the given expression, (5⁻⁸)(5⁻¹⁰), in the coveted form of 5ⁿ. This endeavor will not only solidify our understanding of exponent rules but also enhance our ability to manipulate and simplify mathematical expressions effectively. Let's embark on this mathematical adventure and unravel the intricacies of exponents.
Before we plunge into the specifics of rewriting the expression, it is crucial to establish a firm grasp of the fundamental principles governing exponents. Exponents, in essence, serve as a concise notation for representing repeated multiplication. In the realm of mathematics, an exponent signifies the number of times a base is multiplied by itself. For instance, in the expression 5³, the base is 5, and the exponent is 3, which translates to 5 multiplied by itself three times (5 × 5 × 5), resulting in 125.
Exponents can manifest as positive integers, negative integers, or even fractions, each type possessing unique implications and properties. Positive integer exponents signify repeated multiplication, as demonstrated earlier. Negative integer exponents, on the other hand, denote the reciprocal of the base raised to the corresponding positive exponent. For example, 5⁻² is equivalent to 1/(5²), which equals 1/25. Fractional exponents, also known as rational exponents, represent roots. For instance, 5¹/² signifies the square root of 5.
The product of powers rule is a fundamental principle that governs the multiplication of exponential expressions sharing the same base. This rule asserts that when multiplying expressions with the same base, we can simply add the exponents while retaining the base. Mathematically, this can be expressed as:
aᵐ * aⁿ = aᵐ⁺ⁿ
Where 'a' represents the base, and 'm' and 'n' are the exponents.
This rule stems directly from the definition of exponents as repeated multiplication. When we multiply aᵐ by aⁿ, we are essentially multiplying 'a' by itself 'm' times and then multiplying the result by 'a' multiplied by itself 'n' times. This amalgamation effectively results in 'a' being multiplied by itself a total of 'm + n' times, hence the rule aᵐ * aⁿ = aᵐ⁺ⁿ.
Let's illustrate this rule with a tangible example. Consider the expression 2³ * 2². According to the product of powers rule, we can add the exponents (3 and 2) while retaining the base (2), resulting in 2³⁺² = 2⁵. Evaluating this, we find that 2⁵ equals 32, which aligns with the direct calculation of 2³ * 2² (8 * 4 = 32).
Now, let's wield the power of the product of powers rule to rewrite our given expression, (5⁻⁸)(5⁻¹⁰), in the desired form of 5ⁿ. We can readily discern that both exponential expressions share the same base, which is 5. Therefore, we can directly apply the product of powers rule by adding the exponents.
(5⁻⁸)(5⁻¹⁰) = 5⁻⁸ ⁺ ⁽⁻¹⁰⁾
Simplifying the exponent, we get:
5⁻⁸ ⁺ ⁽⁻¹⁰⁾ = 5⁻¹⁸
Thus, we have successfully rewritten the expression (5⁻⁸)(5⁻¹⁰) in the form 5ⁿ, where n equals -18.
Therefore, the expression (5⁻⁸)(5⁻¹⁰) can be rewritten as:
5⁻¹⁸
This concise representation encapsulates the simplified form of the original expression, adhering to the stipulated form of 5ⁿ.
As we have encountered a negative exponent in our final result (5⁻¹⁸), it is prudent to delve deeper into the nature of negative exponents. Negative exponents, as alluded to earlier, signify the reciprocal of the base raised to the corresponding positive exponent. In essence, a negative exponent indicates division rather than multiplication.
Mathematically, this relationship can be expressed as:
a⁻ⁿ = 1/aⁿ
Where 'a' represents the base, and 'n' is the exponent.
In our specific case, 5⁻¹⁸ can be interpreted as the reciprocal of 5¹⁸. This implies that 5⁻¹⁸ is equivalent to 1 divided by 5 raised to the power of 18.
5⁻¹⁸ = 1/5¹⁸
The magnitude of 5¹⁸ is an exceedingly large number, underscoring the fact that 5⁻¹⁸ represents a minuscule fraction, infinitesimally close to zero.
While the product of powers rule has served us admirably in this endeavor, it is worthwhile to expand our repertoire of exponent rules by introducing the quotient of powers rule. This rule governs the division of exponential expressions sharing the same base. The quotient of powers rule posits that when dividing expressions with the same base, we can subtract the exponents while retaining the base. Mathematically, this can be expressed as:
aᵐ / aⁿ = aᵐ⁻ⁿ
Where 'a' represents the base, and 'm' and 'n' are the exponents.
This rule mirrors the product of powers rule in its rationale. When we divide aᵐ by aⁿ, we are essentially canceling out 'n' factors of 'a' from both the numerator and the denominator, leaving us with 'a' multiplied by itself 'm - n' times, hence the rule aᵐ / aⁿ = aᵐ⁻ⁿ.
For example, consider the expression 3⁵ / 3². According to the quotient of powers rule, we can subtract the exponents (5 and 2) while retaining the base (3), resulting in 3⁵⁻² = 3³. Evaluating this, we find that 3³ equals 27, which aligns with the direct calculation of 3⁵ / 3² (243 / 9 = 27).
In this comprehensive exploration, we have successfully rewritten the expression (5⁻⁸)(5⁻¹⁰) in the form 5ⁿ, where n equals -18. This feat was accomplished by leveraging the product of powers rule, a cornerstone principle governing the multiplication of exponential expressions sharing the same base. Furthermore, we delved into the nature of negative exponents, recognizing their significance in representing reciprocals.
Our journey has not only solidified our understanding of exponent rules but has also honed our ability to manipulate and simplify mathematical expressions effectively. As we continue our mathematical pursuits, the principles and techniques elucidated here will undoubtedly serve as valuable tools in our problem-solving arsenal.
Rewriting exponential expressions can seem daunting at first, but with a clear understanding of the rules and properties involved, it becomes a manageable task. In this guide, we'll break down the process of rewriting the expression (5⁻⁸)(5⁻¹⁰) in the form 5ⁿ, providing a step-by-step approach and explaining the underlying principles. Understanding exponents is crucial for various mathematical concepts, and mastering these rules will greatly enhance your problem-solving skills. This article aims to provide a comprehensive understanding of how to simplify and rewrite exponential expressions, ensuring you grasp the core concepts effectively.
Understanding Exponents: The Foundation of Our Task
Before we dive into the specifics of the given expression, it's important to understand what exponents are and how they work. At its core, an exponent is a way of representing repeated multiplication. For example, 5³ means 5 multiplied by itself three times (5 * 5 * 5). The number being multiplied (in this case, 5) is called the base, and the number indicating how many times to multiply the base (in this case, 3) is called the exponent or power.
Exponents aren't limited to positive integers; they can also be negative, zero, or fractional. Negative exponents represent reciprocals, and fractional exponents represent roots. For instance, 5⁻¹ is the same as 1/5, and 5¹/² is the square root of 5. Understanding these variations is crucial for manipulating and simplifying exponential expressions. Grasping these foundational concepts will make more complex manipulations, like the one we're about to undertake, significantly easier. Make sure you have a solid understanding of the basics before moving forward.
The Product of Powers Rule: Our Key Tool
To rewrite (5⁻⁸)(5⁻¹⁰) in the form 5ⁿ, we'll rely heavily on one of the fundamental rules of exponents: the product of powers rule. This rule states that when multiplying exponential expressions with the same base, you can add the exponents. Mathematically, it's represented as:
aᵐ * aⁿ = aᵐ⁺ⁿ
Where 'a' is the base, and 'm' and 'n' are the exponents. This rule is derived directly from the definition of exponents as repeated multiplication. When you multiply a number raised to one power by the same number raised to another power, you're essentially combining the multiplications. Understanding the product of powers rule is crucial for this and many other exponential simplifications.
For example, consider 2³ * 2². According to the product of powers rule, we can add the exponents: 3 + 2 = 5. So, 2³ * 2² = 2⁵. We can verify this by calculating each side: 2³ is 8, 2² is 4, and 8 * 4 is 32, which is equal to 2⁵. Mastering this rule is a fundamental step in simplifying and rewriting exponential expressions. Now that we've refreshed our understanding of this key rule, let's apply it to our specific problem.
Applying the Product of Powers Rule to (5⁻⁸)(5⁻¹⁰)
Now, let's apply the product of powers rule to our expression, (5⁻⁸)(5⁻¹⁰). Notice that both terms have the same base, which is 5. This means we can directly apply the rule by adding the exponents:
(5⁻⁸)(5⁻¹⁰) = 5⁽⁻⁸⁾ ⁺ ⁽⁻¹⁰⁾
Here, we are adding -8 and -10, which are the exponents of our base, 5. This step is crucial in simplifying the expression and moving closer to the desired form, 5ⁿ. Take careful note of how we handle negative exponents, as they are a common point of confusion. By correctly applying the rule, we transform the multiplication of two exponential terms into a single term with a combined exponent. Now, let’s simplify the exponent and find the value of n.
Simplifying the Exponent: Finding n
Next, we simplify the exponent by adding -8 and -10:
-8 + (-10) = -18
So, our expression becomes:
5⁻¹⁸
This result is now in the form 5ⁿ, where n = -18. Therefore, (5⁻⁸)(5⁻¹⁰) can be rewritten as 5⁻¹⁸. This simple addition is the final step in rewriting the expression. It highlights the power of the product of powers rule in simplifying what might initially seem like a complex expression. Understanding this process allows us to efficiently handle similar problems in the future.
Expressing the Result: 5⁻¹⁸
Therefore, the simplified form of the expression (5⁻⁸)(5⁻¹⁰) in the form 5ⁿ is:
5⁻¹⁸
Here, n = -18. This result clearly demonstrates how the product of powers rule allows us to combine exponential terms with the same base. We’ve successfully transformed a multiplication of two exponential expressions into a single exponential expression. This final result provides a clear and concise answer to the problem, showcasing the power of understanding and applying exponent rules. Make sure to double-check your calculations to confirm the answer.
Understanding Negative Exponents: A Deeper Dive
Since our result includes a negative exponent, it's worth discussing what negative exponents mean in more detail. A negative exponent indicates the reciprocal of the base raised to the positive version of that exponent. In other words:
a⁻ⁿ = 1/aⁿ
So, 5⁻¹⁸ is the same as 1/(5¹⁸). This understanding is crucial for interpreting results that involve negative exponents. It helps to see the connection between exponential expressions and their reciprocal forms. This concept is not just limited to this specific problem but is applicable to a wide range of mathematical situations. Grasping the concept of negative exponents will further enhance your ability to work with exponential expressions.
For example, 2⁻³ is the same as 1/(2³), which equals 1/8. This principle holds true for any base and any negative exponent. Negative exponents are commonly used in scientific notation and in expressing very small numbers. In our case, 5⁻¹⁸ represents an extremely small fraction, as 5¹⁸ is a very large number. Recognizing this relationship between negative exponents and reciprocals is vital for a thorough understanding of exponents.
Practice and Application: Mastering Exponents
Rewriting exponential expressions is a skill that improves with practice. Try working through additional examples to solidify your understanding. For instance, consider expressions like (3⁻²)(3⁻³), (2⁴)(2⁻⁶), or (7⁻¹)(7²). Applying the product of powers rule to these examples will reinforce the concept and build your confidence. The key to mastering exponents is consistent practice and application of the rules. Work through a variety of problems to become proficient.
Moreover, understanding exponents is fundamental in various fields, including science, engineering, and finance. Exponential growth and decay models, scientific notation, and compound interest calculations all rely on a solid grasp of exponents. The more you practice, the more natural these calculations will become. So, take the time to work through several problems and apply what you've learned in different contexts. This will not only help you in academic settings but also in real-world applications.
Conclusion: The Power of Exponent Rules
In this guide, we've demonstrated how to rewrite the expression (5⁻⁸)(5⁻¹⁰) in the form 5ⁿ using the product of powers rule. We've also discussed the meaning of negative exponents and the importance of practice in mastering exponent rules. Understanding these concepts is crucial for success in mathematics and related fields. Exponent rules are powerful tools that, when mastered, can significantly simplify complex expressions. Remember to practice regularly and apply what you've learned to various problems.
By breaking down the problem step-by-step, we've shown that simplifying exponential expressions is an accessible skill. The product of powers rule, along with an understanding of negative exponents, provides a solid foundation for tackling these types of problems. Continue to explore different types of exponential expressions and practice applying the rules. With time and effort, you'll find yourself confidently rewriting and simplifying exponents in a variety of contexts.
Simplifying exponential expressions is a core skill in mathematics, and today, we’re tackling the problem of rewriting (5⁻⁸)(5⁻¹⁰) in the form 5ⁿ. This involves understanding the rules of exponents, particularly how to handle negative exponents and the product of powers rule. This guide breaks down the process into simple steps, providing clear explanations and examples to help you master this essential skill. Our focus is on making the concepts easy to grasp and apply, ensuring you can confidently solve similar problems in the future. Let's dive in and unravel the mystery of exponents!
What are Exponents? A Quick Refresher
Before we jump into the specifics, let's quickly recap what exponents represent. An exponent is a shorthand way of writing repeated multiplication. For example, 5³ means 5 multiplied by itself three times (5 × 5 × 5). The number being multiplied (in this case, 5) is the base, and the number of times it’s multiplied (in this case, 3) is the exponent. Understanding this fundamental definition is the cornerstone for simplifying exponential expressions. Exponents are not just limited to whole numbers; they can be negative, fractional, or even zero, each with its unique implications.
Exponents also play a crucial role in various mathematical concepts, from scientific notation to polynomial expressions. A solid grasp of exponents lays the groundwork for more advanced mathematical topics. Whether dealing with very large or very small numbers, exponents provide a concise and efficient way to represent them. So, make sure you’re comfortable with the basic definition and notation before moving forward.
The Product of Powers Rule: The Key to Simplification
The most important rule for this problem is the product of powers rule. This rule states that when multiplying two exponential expressions with the same base, you can add the exponents. Mathematically, this is expressed as:
aᵐ × aⁿ = aᵐ⁺ⁿ
where 'a' is the base, and 'm' and 'n' are the exponents. This rule is a direct consequence of the definition of exponents as repeated multiplication. When you multiply aᵐ by aⁿ, you’re effectively multiplying 'a' by itself a total of m + n times. Understanding this rule is crucial for simplifying the expression (5⁻⁸)(5⁻¹⁰). It provides a direct path to combining the terms into a single exponential expression.
For instance, consider the example 2² × 2³. According to the product of powers rule, we can add the exponents: 2 + 3 = 5. Therefore, 2² × 2³ = 2⁵. You can verify this by calculating each side: 2² is 4, 2³ is 8, and 4 × 8 is 32, which is equal to 2⁵. Mastering this rule is essential for simplifying and manipulating exponential expressions efficiently. Now, let’s apply this rule to our specific problem.
Applying the Rule to Our Problem: (5⁻⁸)(5⁻¹⁰)
Now, let's use the product of powers rule to simplify (5⁻⁸)(5⁻¹⁰). Notice that both exponential terms have the same base, which is 5. This means we can directly apply the rule and add the exponents:
(5⁻⁸)(5⁻¹⁰) = 5⁽⁻⁸⁾ ⁺ ⁽⁻¹⁰⁾
This step involves combining the exponents -8 and -10, which is a straightforward arithmetic operation. It’s crucial to pay close attention to the signs of the exponents when applying this rule. By correctly applying the product of powers rule, we transform the multiplication of two exponential terms into a single exponential term. This simplification is the core of solving the problem. Next, we'll simplify the exponent itself.
Simplifying the Exponent: Adding -8 and -10
To simplify the exponent, we need to add -8 and -10:
-8 + (-10) = -18
This simple addition gives us the new exponent, -18. Now we can rewrite the expression as:
5⁻¹⁸
This result is now in the desired form, 5ⁿ, where n = -18. The arithmetic operation is straightforward, but it's important to double-check to ensure accuracy. The ability to quickly and correctly add integers, especially negative ones, is vital in simplifying exponents. We have successfully reduced the original expression to its simplest form using the product of powers rule and basic arithmetic.
Expressing the Final Result: 5⁻¹⁸
Thus, the simplified form of (5⁻⁸)(5⁻¹⁰) in the form 5ⁿ is:
5⁻¹⁸
Here, n equals -18. This clear and concise answer highlights the power of the product of powers rule in simplifying exponential expressions. We’ve successfully combined two exponential terms into a single term, making the expression easier to understand and work with. This result demonstrates a fundamental principle in simplifying exponents and serves as a foundation for tackling more complex problems.
Understanding Negative Exponents: What Does 5⁻¹⁸ Mean?
Since our result involves a negative exponent, it’s important to understand what a negative exponent signifies. A negative exponent means we take the reciprocal of the base raised to the positive version of the exponent. In other words:
a⁻ⁿ = 1 / aⁿ
Therefore, 5⁻¹⁸ is the same as 1 / (5¹⁸). This means 5⁻¹⁸ is a very small fraction, since 5¹⁸ is a very large number. This understanding is crucial for interpreting and applying results with negative exponents. Negative exponents are commonly used in scientific notation to represent very small numbers. Recognizing that a negative exponent indicates a reciprocal transformation is key to a comprehensive understanding of exponents.
For example, 2⁻³ is equivalent to 1 / (2³) which simplifies to 1 / 8. This principle applies to any base and any negative exponent. This concept not only helps in simplifying expressions but also provides a deeper understanding of the nature of exponents. Being able to fluidly convert between negative exponents and their reciprocal forms is a valuable skill in mathematics.
Practicing with More Examples: Building Mastery
To truly master simplifying exponential expressions, it’s essential to practice with more examples. Try rewriting the following expressions in the form aⁿ:
- (3⁻³)(3⁻²)
- (2⁵)(2⁻³)
- (7⁻²)(7⁴)
Working through these examples will reinforce your understanding of the product of powers rule and negative exponents. Practice is the key to becoming proficient in mathematics, and exponents are no exception. Each example provides an opportunity to apply the rules and techniques we’ve discussed. As you work through more problems, you’ll develop a more intuitive understanding of how exponents behave.
Furthermore, try varying the problems by including different bases and different exponents, including fractions and decimals. The more varied your practice, the more confident you’ll become in tackling any exponential expression. Don’t hesitate to review the rules and steps as needed, and gradually, you’ll find yourself simplifying exponents with ease.
Conclusion: Mastering Exponents for Mathematical Success
In this comprehensive guide, we’ve demonstrated how to rewrite (5⁻⁸)(5⁻¹⁰) in the form 5ⁿ. We’ve covered the definition of exponents, the product of powers rule, and the meaning of negative exponents. By breaking down the problem into simple steps and providing clear explanations, we’ve aimed to make this topic accessible and understandable. Mastering exponents is a crucial step in your mathematical journey.
The ability to simplify exponential expressions is not only useful in algebra but also in various other fields, including science, engineering, and finance. The key to success is a solid understanding of the rules and consistent practice. Continue to explore different types of exponential expressions and practice applying the rules you’ve learned. With dedication and effort, you’ll master exponents and unlock even more mathematical skills. Keep practicing, and you’ll find yourself simplifying exponents with confidence!
