Simplify (q^8 R^{-2} S^3 / Q R^4 S^{-6})^2: A Step-by-Step Guide

This article provides a detailed, step-by-step solution to simplify the expression (q^8 r^{-2} s^3 / q r^4 s^{-6})2. We will break down the process, explaining each step and the underlying mathematical principles involved. This comprehensive guide aims to help you understand not just the solution, but also the concepts of exponents and how they interact within algebraic expressions. Mastering these concepts is crucial for success in algebra and beyond.

Before diving into the problem, let's quickly review the basic rules of exponents. These rules are the foundation for simplifying complex expressions like the one we have. Understanding these rules will allow us to approach the problem methodically and avoid common pitfalls. Remember, a strong grasp of the fundamentals is key to tackling more advanced problems in mathematics.

• Product of Powers: When multiplying powers with the same base, you add the exponents: x^m * x^n = x^(m+n).
• Quotient of Powers: When dividing powers with the same base, you subtract the exponents: x^m / x^n = x^(m-n).
• Power of a Power: When raising a power to another power, you multiply the exponents: (x^m)n = x^(m*n).
• Negative Exponent: A negative exponent indicates the reciprocal of the base raised to the positive exponent: x^(-n) = 1/x^n.
• Zero Exponent: Any non-zero number raised to the power of zero is 1: x^0 = 1 (where x ≠ 0).

These rules are the fundamental building blocks for simplifying expressions involving exponents. We will be using them extensively in the following steps. Make sure you have a solid understanding of these rules before proceeding.

Now, let's tackle the problem step-by-step. We will apply the rules of exponents we just discussed to simplify the given expression. Each step will be clearly explained to ensure you understand the logic behind the simplification. Remember, the key to solving these types of problems is to break them down into smaller, manageable steps.

H3 Step 1: Simplify the expression inside the parentheses

Our first goal is to simplify the fraction inside the parentheses: (q^8 r^{-2} s^3 / q r^4 s^{-6}). We will use the quotient of powers rule to simplify the terms with the same base. This involves subtracting the exponents of the denominator from the exponents of the numerator. This process will consolidate the terms and make the expression easier to handle.

• For the 'q' terms: q^8 / q^1 = q^(8-1) = q^7
• For the 'r' terms: r^{-2} / r^4 = r^(-2-4) = r^{-6}
• For the 's' terms: s^3 / s^{-6} = s^(3-(-6)) = s^(3+6) = s^9

Combining these simplified terms, we get: q^7 r^{-6} s^9. This simplified expression inside the parentheses is much easier to work with. We have effectively reduced the complexity of the problem by applying the quotient of powers rule.

H3 Step 2: Apply the power of a power rule

Now we have the simplified expression inside the parentheses: q^7 r^-6} s^9**. We need to apply the exponent outside the parentheses, which is 2. This means raising the entire expression to the power of 2 **(q^7 r^{-6 s⁹⁾2. We will use the power of a power rule, which states that (x^m)n = x^(m*n). This rule will be applied to each term within the parentheses.

• For the 'q' term: (q⁷⁾2 = q^(7*2) = q^14
• For the 'r' term: (r^{-6})2 = r^(-6*2) = r^{-12}
• For the 's' term: (s⁹⁾2 = s^(9*2) = s^18

Applying the power of a power rule to each term, we get: q^14 r^{-12} s^18. This further simplifies the expression. We are getting closer to the final answer by systematically applying the rules of exponents.

H3 Step 3: Eliminate the negative exponent

We now have **q^14 r^-12} s^18**. Notice the negative exponent on the 'r' term r^{-12. To express the answer with positive exponents, we need to rewrite this term using the negative exponent rule: x^(-n) = 1/x^n. This means we will move the r^{-12} term to the denominator and change the exponent to positive.

r^{-12} = 1/r^12

So, our expression becomes: q^14 * (1/r^12) * s^18. Combining the terms, we get q^14 s^18 / r^12. This expression now has only positive exponents, which is the desired form of the answer.

H3 Step 4: Write the final answer

Finally, we have simplified the expression to q^14 s^18 / r^12. This is the final answer, expressed with positive exponents. We have successfully applied the rules of exponents to simplify the original expression. This demonstrates the power of breaking down complex problems into smaller, manageable steps.

The simplified expression is:

q^14 s^18 / r^12

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 you arrive at the correct answer. Let's discuss some of these common errors.

• Incorrectly Applying the Quotient of Powers Rule: A common mistake is to subtract the exponents in the wrong order or to apply the rule when the bases are different. Remember, the quotient of powers rule (x^m / x^n = x^(m-n)) only applies when the bases are the same. Always double-check that you are subtracting the exponent in the denominator from the exponent in the numerator.
• Misunderstanding Negative Exponents: Negative exponents can be confusing. Remember that a negative exponent indicates the reciprocal of the base raised to the positive exponent (x^(-n) = 1/x^n). Don't simply change the sign of the exponent; you need to move the term to the denominator (or numerator, if it's already in the denominator).
• Forgetting to Distribute the Power: When raising an expression within parentheses to a power, remember to apply the power to every term inside the parentheses. For example, (xy)^n = x^n y^n. A common mistake is to only apply the power to the first term.
• Adding Exponents When Multiplying Different Bases: The product of powers rule (x^m * x^n = x^(m+n)) only applies when the bases are the same. You cannot add the exponents if the bases are different. For example, x^2 * y^3 cannot be simplified further using this rule.
• Ignoring the Order of Operations: Remember to follow the order of operations (PEMDAS/BODMAS). Exponents should be dealt with before multiplication, division, addition, or subtraction. Failing to follow the correct order can lead to incorrect simplification.

By being mindful of these common mistakes, you can significantly improve your accuracy when simplifying expressions with exponents. Practice and attention to detail are key to mastering these concepts.

To solidify your understanding of simplifying expressions with exponents, it's essential to practice. Here are a few practice problems that you can try. Work through them step-by-step, applying the rules we discussed earlier. The more you practice, the more comfortable you will become with these concepts.

1. Simplify: (a^5 b^{-3} c^2 / a^2 b^4 c^{-1})3
2. Simplify: (x^{-4} y^6 z^0 / x^2 y^{-2} z³⁾-2
3. Simplify: (2p^3 q^{-5} / 8p^{-1} q²⁾2

Working through these practice problems will help you identify any areas where you may need further clarification. Don't be afraid to make mistakes; they are a valuable part of the learning process. Review the steps and rules discussed in this article to guide you through the solutions.

Simplifying expressions with exponents is a fundamental skill in algebra. By understanding the rules of exponents and practicing consistently, you can master this skill. This article has provided a detailed, step-by-step guide to simplifying the expression (q^8 r^{-2} s^3 / q r^4 s^{-6})2. We have also discussed common mistakes to avoid and provided practice problems to help you solidify your understanding. Remember, the key to success is to break down complex problems into smaller, manageable steps and to apply the rules of exponents systematically. Keep practicing, and you will become proficient in simplifying expressions with exponents. This skill will serve you well in your future mathematical endeavors.