Simplifying Expressions With Exponents A Step-by-Step Guide

Exponential expressions can often seem daunting, but with a systematic approach and a firm grasp of the rules of exponents, they can be simplified effectively. The expression $\frac{28 p^9 q^{-5}}{12 p^{-6} q^7}$ is a prime example of such an expression. To simplify this, we'll break it down step by step, leveraging key exponent rules to arrive at the most simplified form. Our journey will involve dividing coefficients, managing variables with exponents in both the numerator and the denominator, and addressing negative exponents to express the final answer with only positive exponents. This process not only simplifies the expression but also reinforces the fundamental principles of manipulating exponents. Remember, the goal is to present the expression in its most concise and understandable form, making it easier to work with in subsequent calculations or applications.

Breaking Down the Expression

To effectively simplify the given expression, $\frac{28 p^9 q^{-5}}{12 p^{-6} q^7}$, it's essential to first isolate the coefficients and the variables. This allows us to apply the rules of exponents more clearly and methodically. The expression can be viewed as a product of three separate parts: the numerical coefficients (28 and 12), the variable p with its exponents (p⁹ and p⁻⁶), and the variable q with its exponents (q⁻⁵ and q⁷). By separating these components, we set the stage for simplifying each part individually before combining them back into a single, simplified expression. This approach not only makes the simplification process more manageable but also highlights how different rules of exponents apply to different parts of the expression.

Simplifying Coefficients

The first step in simplifying the expression is to address the numerical coefficients. We have the fraction ${\frac{28}{12}}$. Both 28 and 12 are divisible by 4. Dividing both the numerator and the denominator by 4, we get:

${ \frac{28}{12} = \frac{28 \div 4}{12 \div 4} = \frac{7}{3} }$

This simplification reduces the numerical part of our expression to a more manageable fraction, which will be easier to work with as we continue to simplify the variable parts. Dealing with coefficients separately is a fundamental technique in simplifying algebraic expressions, ensuring that numerical factors are handled correctly before moving on to variable terms. This approach helps to maintain accuracy and clarity throughout the simplification process.

Handling the Variable p

Now, let's focus on the variable p. We have p⁹ in the numerator and p⁻⁶ in the denominator. According to the quotient rule of exponents, when dividing terms with the same base, we subtract the exponents. Therefore, we have:

${ \frac{p^9}{p^{-6}} = p^{9 - (-6)} = p^{9 + 6} = p^{15} }$

This step demonstrates a crucial rule of exponents: subtracting a negative exponent is equivalent to adding the positive counterpart. The result, p¹⁵, represents a significant simplification of the p terms, making the expression more concise and easier to understand. This manipulation highlights the importance of understanding and applying exponent rules to efficiently simplify algebraic expressions.

Addressing the Variable q

Next, we turn our attention to the variable q. We have q⁻⁵ in the numerator and q⁷ in the denominator. Applying the quotient rule of exponents, we subtract the exponents:

${ \frac{q^{-5}}{q^7} = q^{-5 - 7} = q^{-12} }$

However, we want to express our final answer with positive exponents only. To achieve this, we recall that a term with a negative exponent in the numerator can be moved to the denominator with a positive exponent (and vice versa). Therefore:

${ q^{-12} = \frac{1}{q^{12}} }$

This step is crucial for ensuring that the simplified expression adheres to the convention of using positive exponents. By moving q⁻¹² to the denominator as q¹², we transform the expression into a more standard and easily interpretable form. This manipulation underscores the flexibility and power of exponent rules in rewriting expressions to meet specific requirements.

Combining Simplified Components

After simplifying the coefficients, the variable p terms, and the variable q terms, we can now combine these simplified components back into a single expression. This involves bringing together the simplified numerical fraction and the simplified variable terms to form the final, most concise representation of the original expression. The process of combining these parts is a crucial step in ensuring that the entire expression is correctly represented in its simplest form.

Putting It All Together

We found that the simplified coefficient is ${\frac{7}{3}}$, the simplified p term is p¹⁵, and the simplified q term is ${\frac{1}{q^{12}}}$. Combining these, we get:

${ \frac{7}{3} \cdot p^{15} \cdot \frac{1}{q^{12}} = \frac{7p^{15}}{3q^{12}} }$

This result represents the fully simplified form of the original expression. By multiplying the simplified components together, we have successfully consolidated the numerical and variable terms into a single fraction, demonstrating the power of breaking down complex expressions into manageable parts. This final combination step highlights the importance of precision and attention to detail in ensuring that the simplified expression accurately reflects the original problem.

Final Answer

Thus, the expression $\frac{28 p^9 q^{-5}}{12 p^{-6} q^7}$ simplifies to $\frac{7 p^{15}}{3 q^{12}}$.

Selecting the Correct Option

Comparing our simplified expression to the given options:

• A. $\frac{2}{p^{15} q^{12}}$
• B. $\frac{7 p^{15}}{3 q^{12}}$
• C. $\frac{2 q^{12}}{p^{15}}$
• D. $\frac{7 p^{15} q^{12}}{3}$

We can clearly see that option B, $\frac{7 p^{15}}{3 q^{12}}$, matches our simplified expression. This step ensures that we accurately identify the correct answer from the provided choices, reinforcing the importance of careful comparison and verification in problem-solving.

Conclusion

In conclusion, simplifying expressions with exponents requires a systematic approach and a solid understanding of exponent rules. By breaking down the expression into smaller parts, simplifying each part individually, and then combining the results, we can effectively simplify complex expressions. The expression $\frac{28 p^9 q^{-5}}{12 p^{-6} q^7}$ serves as an excellent example of this process, highlighting the importance of simplifying coefficients, applying the quotient rule, and handling negative exponents. This comprehensive approach not only leads to the correct simplified form but also enhances one's ability to tackle similar problems with confidence and accuracy.

Mastering Exponent Rules

Mastering exponent rules is crucial for success in algebra and beyond. This example demonstrates the power of breaking down a complex problem into manageable steps. Remember to simplify coefficients, apply the quotient rule for exponents, and handle negative exponents with care. With practice, you'll become more comfortable and confident in your ability to simplify exponential expressions. This mastery will not only help you in academic settings but also in various real-world applications where mathematical expressions need to be simplified for clarity and efficiency. Continuous practice and application of these rules are key to developing a strong foundation in algebra.

The Importance of Simplification

Simplification is a fundamental skill in mathematics. It allows us to express complex ideas in a clear and concise manner. Whether you're working on algebraic equations, calculus problems, or even real-world applications, the ability to simplify expressions is invaluable. It not only makes the problem easier to solve but also helps in understanding the underlying concepts more effectively. This skill is not just about getting the correct answer; it's about developing a deeper understanding of mathematical structures and relationships. Therefore, investing time and effort in mastering simplification techniques is a worthwhile endeavor for any student of mathematics.

Practice Makes Perfect

Practice is key to mastering any mathematical concept, and simplifying expressions with exponents is no exception. Work through a variety of problems, starting with simpler ones and gradually increasing the difficulty. Pay close attention to the steps involved and the rules you are applying. The more you practice, the more natural these steps will become, and the more confident you will be in your ability to tackle complex problems. Remember, each problem you solve is an opportunity to reinforce your understanding and improve your skills. So, embrace the challenge, and keep practicing!