Simplifying Expressions: (4p^5q^-10) / (2p^-2q^5)

Oct 29, 2025 by Jhon Lennon 50 views

Alright guys, let's break down how to simplify the expression (4p⁵q⁻¹⁰) / (2p⁻²q⁵). This looks a bit intimidating at first, but don't worry, we'll get through it step by step. Simplifying algebraic expressions like this involves applying the rules of exponents and basic arithmetic. By understanding these rules, we can make complex expressions much easier to work with. Remember, the goal is to combine like terms and reduce the expression to its simplest form. So, grab your pencils, and let's dive in!

Understanding the Basics

Before we jump into the main problem, let’s quickly recap some fundamental rules of exponents that we'll be using. These rules are the bread and butter of simplifying expressions like this, and knowing them well will make the whole process much smoother. Trust me, once you've got these down, you'll be simplifying expressions like a pro!

Division of Like Bases: When dividing terms with the same base, you subtract the exponents. Mathematically, this is expressed as aˣ / aʸ = aˣ⁻ʸ. This rule is super important because it allows us to combine terms that might look separate at first glance. For example, if you have p⁵ / p², you simply subtract the exponents to get p³.
Negative Exponents: A term with a negative exponent can be rewritten as its reciprocal with a positive exponent. That is, a⁻ˣ = 1 / aˣ. Negative exponents might seem a bit weird, but they're actually quite useful. They allow us to move terms from the numerator to the denominator (or vice versa) to make the expression easier to manage. For instance, q⁻¹⁰ becomes 1 / q¹⁰.
Coefficient Division: When you have coefficients (the numbers in front of the variables), you simply divide them as you normally would. For example, 4 / 2 = 2. This is just basic arithmetic, but it's an essential part of simplifying the entire expression.
Combining Like Terms: Always remember to combine like terms whenever possible. This means adding or subtracting coefficients of terms that have the same variable and exponent. For example, 3p² + 5p² = 8p². Combining like terms helps to streamline the expression and make it as simple as possible.

With these rules in mind, we can tackle the expression step by step, making sure to apply each rule correctly. This isn't just about getting the right answer; it's about understanding the process so you can apply these skills to other problems in the future. So, let's keep these basics in our toolbox as we move forward!

Step-by-Step Simplification

Now that we've got our tools ready, let's get to work on simplifying the expression (4p⁵q⁻¹⁰) / (2p⁻²q⁵). We'll take it one step at a time to make sure everything is clear and easy to follow. Remember, patience is key when dealing with these kinds of problems!

Divide the Coefficients: First, let's deal with the coefficients. We have 4 in the numerator and 2 in the denominator. Dividing these gives us: 4 / 2 = 2 So, our expression now starts with 2.
Simplify the 'p' Terms: Next, let's simplify the terms involving 'p'. We have p⁵ in the numerator and p⁻² in the denominator. Using the rule for dividing like bases, we subtract the exponents: p⁵ / p⁻² = p⁵⁻⁽⁻²⁾ = p⁵⁺² = p⁷ So, we now have 2p⁷.
Simplify the 'q' Terms: Now, let's move on to the 'q' terms. We have q⁻¹⁰ in the numerator and q⁵ in the denominator. Again, we subtract the exponents: q⁻¹⁰ / q⁵ = q⁻¹⁰⁻⁵ = q⁻¹⁵ So, we now have 2p⁷q⁻¹⁵.
Handle the Negative Exponent: We have a negative exponent for 'q', which isn't usually considered simplified. To get rid of the negative exponent, we move q⁻¹⁵ to the denominator and make the exponent positive: q⁻¹⁵ = 1 / q¹⁵ So, 2p⁷q⁻¹⁵ becomes 2p⁷ / q¹⁵.
Final Simplified Expression: Putting it all together, the simplified expression is: 2p⁷ / q¹⁵ And that's it! We've successfully simplified the expression by applying the rules of exponents and basic arithmetic. Remember, the key is to take it one step at a time and stay organized.

Detailed Breakdown of Exponent Rules

To truly master simplifying expressions like (4p⁵q⁻¹⁰) / (2p⁻²q⁵), it's crucial to have a solid understanding of exponent rules. These rules aren't just abstract concepts; they are the tools we use to manipulate and simplify algebraic expressions. Let's dive deeper into these rules to ensure you're comfortable using them.

Product of Powers Rule: When you multiply two terms with the same base, you add their exponents. Mathematically, this is represented as aˣ * aʸ = aˣ⁺ʸ. This rule is fundamental because it allows you to combine terms that might initially appear separate. For example, if you have p³ * p⁴, you simply add the exponents to get p⁷. Understanding this rule is essential for simplifying more complex expressions.
Quotient of Powers Rule: As we used in the main problem, when you divide two terms with the same base, you subtract the exponents. This is expressed as aˣ / aʸ = aˣ⁻ʸ. This rule is the counterpart to the product of powers rule and is equally important. For instance, if you have p⁸ / p², you subtract the exponents to get p⁶. Knowing when to apply this rule can significantly simplify your calculations.
Power of a Power Rule: When you raise a power to another power, you multiply the exponents. This is represented as (aˣ)ʸ = aˣʸ. This rule is useful when dealing with expressions that have exponents nested within exponents. For example, if you have (p²)³, you multiply the exponents to get p⁶. Mastering this rule will help you tackle more intricate problems with ease.
Power of a Product Rule: When you raise a product to a power, you apply the power to each factor in the product. This is expressed as (ab)ˣ = aˣbˣ. This rule is particularly helpful when dealing with expressions that have multiple variables inside parentheses. For example, if you have (2p)³, you apply the exponent to both the 2 and the p to get 8p³.
Power of a Quotient Rule: When you raise a quotient to a power, you apply the power to both the numerator and the denominator. This is expressed as (a/b)ˣ = aˣ / bˣ. This rule is similar to the power of a product rule but applies to division. For example, if you have (p/q)², you apply the exponent to both the p and the q to get p² / q².
Zero Exponent Rule: Any non-zero number raised to the power of zero is equal to 1. This is expressed as a⁰ = 1 (where a ≠ 0). This rule might seem odd, but it's a fundamental part of the exponent rules. For example, p⁰ = 1, regardless of what p is (as long as it's not zero).
Negative Exponent Rule: A term with a negative exponent can be rewritten as its reciprocal with a positive exponent. That is, a⁻ˣ = 1 / aˣ. We used this rule in the main problem to handle q⁻¹⁵. Negative exponents allow you to move terms between the numerator and denominator, which can be very useful in simplifying expressions.
Fractional Exponents Rule: A fractional exponent indicates a root. For example, a^(1/n) is the nth root of a. More generally, a^(m/n) is the nth root of a raised to the mth power. Understanding fractional exponents allows you to work with roots and exponents in a unified way.

By understanding these exponent rules in detail, you'll be well-equipped to tackle a wide range of simplification problems. Practice applying these rules in different scenarios to build your confidence and skills. Remember, the key is to break down complex problems into smaller, manageable steps and apply the appropriate rules along the way.

Common Mistakes to Avoid

When simplifying expressions, it's easy to make common mistakes that can throw off your entire solution. Being aware of these pitfalls can help you avoid them and ensure you get the correct answer. Let's go over some of the most frequent errors and how to steer clear of them.

Incorrectly Applying the Division Rule: One common mistake is to add exponents when dividing terms with the same base, instead of subtracting them. Remember, when you divide aˣ by aʸ, you should subtract the exponents (aˣ⁻ʸ), not add them. For example, p⁵ / p² should be p³, not p⁷.
Misunderstanding Negative Exponents: Negative exponents often cause confusion. Remember that a negative exponent means you should take the reciprocal of the base raised to the positive exponent. For example, p⁻² is 1 / p², not -p². It's crucial to understand this to avoid flipping the term incorrectly.
Forgetting to Apply the Exponent to All Factors: When raising a product to a power, make sure to apply the exponent to all factors within the parentheses. For example, (2p)³ should be 8p³, not 2p³. Failing to apply the exponent to all factors is a common oversight.
Not Simplifying Coefficients Correctly: Sometimes, people forget to simplify the coefficients correctly. Always divide or multiply the coefficients as you normally would. For example, if you have (6p²) / (3p), make sure to divide 6 by 3 to get 2.
Mixing Up Addition and Multiplication of Exponents: It's easy to mix up when to add exponents and when to multiply them. Remember, you add exponents when multiplying terms with the same base (aˣ * aʸ = aˣ⁺ʸ) and multiply exponents when raising a power to another power ((aˣ)ʸ = aˣʸ). Keeping these rules separate is essential.
Ignoring the Order of Operations: Always follow the order of operations (PEMDAS/BODMAS) to ensure you're performing the correct operations in the right order. This is especially important when dealing with more complex expressions.
Not Combining Like Terms: Failing to combine like terms can leave your expression unsimplified. Always look for terms with the same variable and exponent and combine them. For example, 3p² + 5p² should be simplified to 8p².
Incorrectly Handling Zero Exponents: Remember that any non-zero number raised to the power of zero is 1. For example, p⁰ = 1 (as long as p ≠ 0). Misunderstanding this rule can lead to incorrect simplifications.

By being mindful of these common mistakes and taking the time to double-check your work, you can improve your accuracy and confidence in simplifying expressions. Remember, practice makes perfect, so keep working on these skills to master them!

Practice Problems

To really nail down the concepts we've covered, let's work through a few practice problems. These exercises will help you apply the rules and techniques we've discussed, solidifying your understanding and boosting your confidence. Grab a pen and paper, and let's get started!

Problem 1: Simplify (6a³b⁻²) / (3a⁻¹b⁴)

Solution:

Divide the coefficients: 6 / 3 = 2
Simplify the 'a' terms: a³ / a⁻¹ = a³⁻⁽⁻¹⁾ = a⁴
Simplify the 'b' terms: b⁻² / b⁴ = b⁻²⁻⁴ = b⁻⁶
Handle the negative exponent: b⁻⁶ = 1 / b⁶
Final simplified expression: 2a⁴ / b⁶

Problem 2: Simplify (5x⁴y²) * (2x⁻²y⁻³)

Solution:

Multiply the coefficients: 5 * 2 = 10
Simplify the 'x' terms: x⁴ * x⁻² = x⁴⁺⁽⁻²⁾ = x²
Simplify the 'y' terms: y² * y⁻³ = y²⁺⁽⁻³⁾ = y⁻¹
Handle the negative exponent: y⁻¹ = 1 / y
Final simplified expression: 10x² / y

Problem 3: Simplify (4p²q³)⁻²

Solution:

Apply the power to all factors: (4p²q³)⁻² = 4⁻² * (p²)⁻² * (q³)⁻²
Simplify each term: 4⁻² = 1 / 4² = 1 / 16, (p²)⁻² = p⁻⁴, (q³)⁻² = q⁻⁶
Handle the negative exponents: p⁻⁴ = 1 / p⁴, q⁻⁶ = 1 / q⁶
Final simplified expression: 1 / (16p⁴q⁶)

Problem 4: Simplify √ (9a⁴b⁶)

Solution:

Rewrite the square root as a fractional exponent: √ (9a⁴b⁶) = (9a⁴b⁶)^(1/2)
Apply the exponent to all factors: 9^(1/2) * (a⁴)^(1/2) * (b⁶)^(1/2)
Simplify each term: 9^(1/2) = 3, (a⁴)^(1/2) = a², (b⁶)^(1/2) = b³
Final simplified expression: 3a²b³

Problem 5: Simplify (12x⁵y⁻³) / (4x²y⁻⁵)

Solution:

Divide the coefficients: 12 / 4 = 3
Simplify the 'x' terms: x⁵ / x² = x⁵⁻² = x³
Simplify the 'y' terms: y⁻³ / y⁻⁵ = y⁻³⁻⁽⁻⁵⁾ = y²
Final simplified expression: 3x³y²

These practice problems cover a range of scenarios you might encounter when simplifying expressions. Work through them carefully, and don't hesitate to review the rules and techniques we've discussed if you get stuck. The more you practice, the more comfortable and confident you'll become!

Conclusion

So, there you have it, guys! Simplifying expressions like (4p⁵q⁻¹⁰) / (2p⁻²q⁵) might seem daunting at first, but with a solid understanding of the basic rules and a step-by-step approach, it becomes much more manageable. We've covered the key exponent rules, common mistakes to avoid, and worked through several practice problems to help you master these skills. Remember, the key is to break down complex problems into smaller, more manageable steps and to stay organized. Keep practicing, and you'll be simplifying expressions like a pro in no time!