Simplifying (4x^3y^5)(3x^5y)^2 A Step By Step Guide

Jul 9, 2025 by qnaunigon 52 views

Simplifying Expressions with Exponents: A Comprehensive Guide

In the realm of algebra, simplifying expressions is a fundamental skill. When dealing with expressions involving exponents, it's crucial to understand and apply the rules of exponents correctly. This article aims to provide a comprehensive guide on how to simplify algebraic expressions with exponents, focusing on the expression $\left(4 x^3 y^5\right)\left(3 x^5 y\right)^2$ as an illustrative example. We will break down the simplification process step-by-step, ensuring clarity and understanding for readers of all backgrounds. Mastering these techniques is essential for success in algebra and higher-level mathematics.

Understanding the Expression

Before we dive into the simplification process, let's first dissect the expression $\left(4 x^3 y^5\right)\left(3 x^5 y\right)^2$ . This expression involves several components: coefficients (the numerical parts), variables (represented by letters), and exponents (the superscripts indicating the power to which a base is raised). To simplify this expression effectively, we need to apply the rules of exponents and the order of operations (PEMDAS/BODMAS). The expression consists of two terms multiplied together. The first term, $4x^3y^5$ , has a coefficient of 4, variables $x$ and $y$ , and exponents 3 and 5 for $x$ and $y$ , respectively. The second term, $\left(3 x^5 y\right)^2$ , is a bit more complex as it involves an exponent outside the parentheses. This means we need to apply the power of a product rule before we can multiply the terms together. Understanding the structure of the expression is the first key step in simplifying it. We will now delve into the specific rules of exponents that will guide us through the simplification process, ensuring a clear and methodical approach to solving the problem. Grasping these foundational concepts is crucial for anyone looking to excel in algebra and related mathematical fields.

Rules of Exponents

The rules of exponents are the bedrock of simplifying expressions like the one we're tackling. Let's review the key rules that will be instrumental in our simplification process:

Power of a Product Rule: This rule states that when a product is raised to a power, each factor in the product is raised to that power. Mathematically, it's expressed as $(ab)^n = a^n b^n$ . This rule is critical for dealing with terms like $(3x^5y)^2$ in our expression.
Power of a Power Rule: When a power is raised to another power, you multiply the exponents. This is represented as $(a^m)^n = a^{mn}$ . This rule will come into play when we apply the power of a product rule.
Product of Powers Rule: When multiplying terms with the same base, you add the exponents. This rule is expressed as $a^m * a^n = a^{m+n}$ . This rule will be essential when we combine the terms after applying the power of a product rule.

These three rules are the primary tools we'll use to simplify the expression. Understanding and applying these rules correctly is essential for achieving the correct result. Let's illustrate these rules with simple examples before we apply them to our main expression. For instance, using the power of a product rule, $(2x)^3 = 2^3 * x^3 = 8x^3$ . Using the power of a power rule, $(x^2)^3 = x^(2*3) = x^6$ . And using the product of powers rule, $x^2 * x^3 = x^(2+3) = x^5$ . With these rules firmly in mind, we can now confidently proceed to simplify the given expression, breaking it down into manageable steps for clarity and accuracy.

Step-by-Step Simplification

Now, let's apply these rules of exponents to simplify the expression $\left(4 x^3 y^5\right)\left(3 x^5 y\right)^2$ step-by-step. This methodical approach ensures we don't miss any crucial details and arrive at the correct simplified form.

Step 1: Apply the Power of a Product Rule

The first step involves simplifying the term $\left(3 x^5 y\right)^2$ . According to the power of a product rule, we need to raise each factor inside the parentheses to the power of 2. This gives us:

$\left(3 x^5 y\right)^2 = 3^2 * \left(x^5\right)^2 * y^2$

Now, we can simplify further:

$3^2 = 9$

Applying the power of a power rule to $\left(x^5\right)^2$ , we multiply the exponents:

$\left(x^5\right)^2 = x^{5*2} = x^{10}$

So, the term $\left(3 x^5 y\right)^2$ simplifies to:

$9x^{10}y^2$

This step is crucial as it eliminates the parentheses and sets the stage for combining like terms. By breaking down the process into smaller, manageable steps, we minimize the chances of making errors. The careful application of the power of a product rule is the foundation for the subsequent steps in the simplification process. Now that we've simplified the second term, we can move on to the next step, which involves multiplying the simplified term with the first term in the expression.

Step 2: Multiply the Terms

Now that we've simplified $\left(3 x^5 y\right)^2$ to $9x^{10}y^2$ , we can multiply it with the first term, $\left(4 x^3 y^5\right)$ . This step involves combining the coefficients and applying the product of powers rule to the variables. The expression now looks like this:

$\left(4 x^3 y^5\right) * \left(9 x^{10} y^2\right)$

First, let's multiply the coefficients:

$4 * 9 = 36$

Next, we'll multiply the terms with the same base. For the variable $x$ , we have $x^3$ and $x^{10}$ . Applying the product of powers rule, we add the exponents:

$x^3 * x^{10} = x^{3+10} = x^{13}$

Similarly, for the variable $y$ , we have $y^5$ and $y^2$ . Applying the product of powers rule again:

$y^5 * y^2 = y^{5+2} = y^7$

Now, we combine the results:

$36x^{13}y^7$

This step demonstrates the power of the product of powers rule in simplifying expressions. By systematically combining like terms, we reduce the expression to its simplest form. The careful multiplication of coefficients and the addition of exponents are key to achieving the correct result. With this step completed, we have successfully simplified the expression, arriving at the final answer.

Final Simplified Expression

After applying the rules of exponents and performing the necessary multiplications, we arrive at the final simplified expression. By meticulously following each step, we've transformed the original complex expression into a more concise and manageable form. The final simplified expression is:

$36x^{13}y^7$

This result is the equivalent of the original expression $\left(4 x^3 y^5\right)\left(3 x^5 y\right)^2$ , but it is now in its simplest form. This means that there are no more like terms to combine and no more exponents to simplify. The expression is now ready for further use in algebraic manipulations or problem-solving scenarios. Understanding how to arrive at this simplified form is a crucial skill in algebra. It allows us to work with expressions more efficiently and accurately. Moreover, it provides a solid foundation for tackling more complex algebraic problems. The process we've followed, from understanding the expression to applying the rules of exponents and simplifying step-by-step, is a valuable template for simplifying other algebraic expressions. This mastery of simplification techniques is a key element in achieving success in mathematics.

Common Mistakes to Avoid

While simplifying expressions with exponents, it's easy to make mistakes if you're not careful. Recognizing these common pitfalls can help you avoid errors and ensure accurate results. Here are some common mistakes to watch out for:

Forgetting to Apply the Power to All Factors: When using the power of a product rule, remember to apply the exponent to every factor inside the parentheses. For example, $\left(3 x^2\right)^2$ is $9x^4$ , not $3x^4$ . The coefficient must also be raised to the power.
Incorrectly Adding Exponents: The product of powers rule ( $a^m * a^n = a^{m+n}$ ) only applies when multiplying terms with the same base. Don't add exponents when the bases are different or when adding or subtracting terms.
Misunderstanding the Power of a Power Rule: When raising a power to another power, you multiply the exponents, not add them. For example, $\left(x^3\right)^2 = x^6$ , not $x^5$ .
Ignoring the Order of Operations: Always follow the order of operations (PEMDAS/BODMAS). Exponents should be dealt with before multiplication or division.
Mistakes with Negative Exponents: Remember that a negative exponent indicates a reciprocal. For example, $x^{-2} = \frac{1}{x^2}$ . Failing to handle negative exponents correctly can lead to significant errors.

By being aware of these common mistakes, you can approach simplification problems with greater confidence and accuracy. It's always a good practice to double-check your work, especially when dealing with exponents. Paying close attention to the details and applying the rules of exponents correctly are essential for achieving success in algebra.

Practice Problems

To solidify your understanding of simplifying expressions with exponents, working through practice problems is essential. These exercises will help you apply the rules we've discussed and identify any areas where you may need further clarification. Here are a few practice problems to get you started:

Simplify: $\left(2a^2b^3\right)\left(5a^4b\right)^2$
Simplify: $\frac{12x^5y^2}{4x^2y^6}$
Simplify: $\left(\frac{3m^4}{n^2}\right)^3$
Simplify: $\left(4p^{-2}q^3\right)\left(2p^5q^{-1}\right)$
Simplify: $\sqrt{16x^6y^8}$

For each problem, try to follow the step-by-step approach we've outlined in this article. Remember to apply the rules of exponents carefully and double-check your work. If you encounter any difficulties, revisit the relevant sections of this guide or seek additional resources. The key to mastering simplification is consistent practice and a thorough understanding of the rules. By tackling a variety of problems, you'll build confidence in your ability to simplify expressions with exponents accurately and efficiently. Remember, practice makes perfect, and the more you work with these concepts, the more natural they will become.

Conclusion

In conclusion, simplifying algebraic expressions with exponents is a fundamental skill in algebra. By understanding and applying the rules of exponents, such as the power of a product rule, power of a power rule, and product of powers rule, we can effectively reduce complex expressions to their simplest forms. We have demonstrated this process through a detailed, step-by-step simplification of the expression $\left(4 x^3 y^5\right)\left(3 x^5 y\right)^2$ , arriving at the simplified form $36x^{13}y^7$ . Additionally, we have highlighted common mistakes to avoid and provided practice problems to reinforce your understanding.

Mastering these techniques is not only crucial for success in algebra but also serves as a foundation for more advanced mathematical concepts. The ability to manipulate and simplify expressions efficiently is a valuable asset in various fields of mathematics, science, and engineering. By consistently practicing and applying the principles outlined in this guide, you can develop a strong foundation in algebra and confidently tackle a wide range of mathematical challenges. Remember, the key to success lies in understanding the underlying rules, practicing regularly, and paying attention to detail. With dedication and perseverance, you can master the art of simplifying expressions with exponents and unlock new levels of mathematical proficiency.