Simplify Algebraic Expressions With Exponents

When you're diving into the world of algebra, you'll often encounter expressions that look a bit daunting at first glance. Take, for instance, the expression: r⁵ ÷ r⁴ × r² ÷ r⁵. This might seem like a complex puzzle, but with a solid understanding of exponent rules, it becomes incredibly straightforward to simplify. We're going to break down how to tackle this problem step-by-step, making sure you feel confident in simplifying similar expressions. The key to mastering these problems lies in remembering the fundamental rules of exponents, especially when dealing with multiplication and division.

Understanding the Rules of Exponents

Before we jump into simplifying our specific expression, let's quickly recap the essential exponent rules that will be our best friends. These rules are the bedrock upon which all exponent simplification is built. The first rule we'll use is for division: when dividing exponents with the same base, you subtract the powers. So, xᵃ / xᵇ = xᵃ⁻ᵇ. This rule is crucial because our expression involves division. Think of it as combining terms by reducing their powers. The second rule we'll leverage is for multiplication: when multiplying exponents with the same base, you add the powers. Thus, xᵃ × xᵇ = xᵃ⁺ᵇ. This rule comes into play when we combine terms that are being multiplied. It's the inverse operation of division in terms of exponents. Knowing these two rules inside out will enable you to navigate through any expression involving multiplication and division of terms with the same base. It's like having a secret code to unlock the simplified form of these expressions. For example, if you see x³ × x⁵, you simply add the exponents to get x⁸. Conversely, if you see x⁹ / x², you subtract the exponents to get x⁷. These operations are fundamental and appear in countless algebraic scenarios. Mastering them now will save you a lot of time and confusion later on. We'll be applying these rules directly to our problem, showing how they work in practice to unravel the complexity of the given expression. Remember, the base 'r' remains the same throughout these operations; only the exponents change according to the rules.

Step-by-Step Simplification

Now, let's put those rules into action and simplify the expression r⁵ ÷ r⁴ × r² ÷ r⁵. We'll tackle this from left to right, applying the rules as we go. First, consider the division part: r⁵ ÷ r⁴. Using the division rule (subtracting exponents), this simplifies to r⁵⁻⁴ = r¹. So, our expression now looks like: r¹ × r² ÷ r⁵. Next, we have a multiplication: r¹ × r². Applying the multiplication rule (adding exponents), we get r¹⁺² = r³. Our expression is now reduced to: r³ ÷ r⁵. Finally, we have one last division: r³ ÷ r⁵. Again, using the division rule, we subtract the exponents: r³⁻⁵ = r⁻². So, the simplified form of the expression is r⁻². This is a perfectly valid simplified form. However, in many contexts, it's preferred to express answers with positive exponents. To convert a negative exponent to a positive one, we use the rule x⁻ⁿ = 1 / xⁿ. Therefore, r⁻² can be rewritten as 1 / r². This final form, 1 / r², is often the most desired answer when simplifying expressions involving negative exponents. It clearly shows the relationship between the variable and its power in a positive, understandable way. Each step logically follows from the previous one, driven by the consistent application of exponent laws. The journey from the initial complex-looking expression to the final simple form 1 / r² demonstrates the power and elegance of these algebraic rules.

Evaluating the Options

We've successfully simplified the expression to 1 / r². Now, let's look at the provided options to see which one matches our result. We need to find the option that is equivalent to 1 / r². Let's analyze each one:

(A) r² r³: This option involves multiplying two terms with the same base. Using the multiplication rule, we add the exponents: r²⁺³ = r⁵. This is not equal to 1 / r².

(B) r² ÷ r²: This option involves dividing a term by itself. Using the division rule, we subtract the exponents: r²⁻² = r⁰. Any non-zero number raised to the power of zero is 1. So, this simplifies to 1. This is also not equal to 1 / r².

(C) 1 ÷ (r¹⁵ r⁴): Let's first simplify the part inside the parentheses. r¹⁵ × r⁴ means adding the exponents: r¹⁵⁺⁴ = r¹⁹. So, this option becomes 1 ÷ r¹⁹, or 1 / r¹⁹. This is not our answer.

(D) 1 ÷ (r² r³): First, simplify the expression within the parentheses: r² × r³. Using the multiplication rule, we add the exponents: r²⁺³ = r⁵. So, this option becomes 1 ÷ r⁵, or 1 / r⁵. This is also not our answer.

Wait a minute! It seems there might be a slight discrepancy between our derived answer and the provided options. Let's re-examine our simplification process and the options very carefully. Sometimes, a simple transcription error or a misunderstanding of how options are presented can lead to confusion. Our step-by-step derivation led us unequivocally to 1 / r². Let's assume for a moment that one of the options should represent 1 / r² and re-evaluate. If we look closely at the structure, perhaps option (D) was intended to be different. However, based strictly on what's written, none of the options precisely match 1 / r². This highlights the importance of accurate problem statement and option formulation in mathematics. In a real test scenario, if this happened, you would double-check your work and then consider if there was a typo in the question or options. Sometimes, the closest answer might be considered, but that's not ideal. Let's assume there was a typo in option (D) and it was meant to be **1 ÷ (r² ÷ r³) ** or similar. If it were **1 ÷ (r² ÷ r³) **, that would be 1 ÷ r²⁻³ = 1 ÷ r⁻¹ = 1 ÷ (1/r) = r. Still not right. What if it was 1 ÷ (r³ ÷ r⁵)? That would be 1 ÷ r³⁻⁵ = 1 ÷ r⁻² = 1 / (1/r²) = r². Still not 1/r². It seems there might be an error in the provided options. However, let's consider the possibility of a mistake in my calculation, though the exponent rules are quite standard. Let's retrace: r⁵ ÷ r⁴ × r² ÷ r⁵. Grouping divisions and multiplications can sometimes help if order of operations is ambiguous without parentheses, but typically it's left to right. (r⁵ / r⁴) * r² / r⁵. This is (r⁵⁻⁴) * r² / r⁵ = r¹ * r² / r⁵ = r¹⁺² / r⁵ = r³ / r⁵ = r³⁻⁵ = r⁻² = 1/r². The calculation seems solid. Let's double check the options one last time, assuming the question is correct. Perhaps I misinterpreted the notation in the options. 'r² r³' usually means r² * r³. 'r² ÷ r²' is clear. '1 ÷ (r¹⁵ r⁴)' is clear. '1 ÷ (r² r³)' is clear. Given our confirmed simplification to 1 / r², and carefully reviewing the options, it appears there's an issue with the provided choices. In a situation like this, one might indicate that none of the options are correct or seek clarification.

Conclusion and Further Exploration

We've diligently worked through the expression r⁵ ÷ r⁴ × r² ÷ r⁵ using the fundamental rules of exponents. Our step-by-step simplification led us to the result r⁻², which is equivalent to 1 / r². This process involved applying the rule for dividing exponents with the same base (subtracting powers) and the rule for multiplying exponents with the same base (adding powers). The final step of converting a negative exponent to a positive one is also a key technique. Although none of the provided options perfectly matched our derived answer 1 / r², this exercise underscores the importance of understanding these core algebraic principles. Mastering exponent rules is not just about solving textbook problems; it's a foundational skill that empowers you to simplify complex mathematical expressions encountered in calculus, physics, engineering, and beyond. The ability to manipulate exponents efficiently can significantly streamline calculations and deepen your understanding of mathematical relationships. For instance, understanding how exponents work is critical when dealing with scientific notation, compound interest calculations, or analyzing the growth rates of functions.

If you're interested in further refining your skills in algebra and simplifying expressions, I highly recommend exploring resources that offer practice problems and in-depth explanations. Understanding algebraic manipulation is a continuous journey, and consistent practice is key. For more detailed information on exponent rules and algebraic simplification, you can refer to reputable educational websites.

• Khan Academy offers a comprehensive range of free courses and practice exercises on algebra, including detailed lessons on exponents. You can find them by searching for "Khan Academy algebra exponents" on your preferred search engine.
• The Math is Fun website provides clear, easy-to-understand explanations of mathematical concepts, including a dedicated section on exponents with helpful examples. You can search for "Math is Fun exponents" to access their resources.