Simplifying Exponential Expressions A Step-by-Step Guide

In the realm of mathematics, simplifying expressions is a fundamental skill. This guide delves into the process of simplifying exponential expressions, focusing on a specific example: (2^9 × 3^5) × (2^4 × 3)^2. We'll break down the steps, explain the underlying principles, and provide a comprehensive understanding of how to approach such problems. Mastering these techniques will empower you to tackle more complex mathematical challenges with confidence.

Understanding Exponential Notation

Before we dive into the simplification process, let's revisit the basics of exponential notation. An expression like 2^9 represents 2 multiplied by itself 9 times (2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2). The number 2 is the base, and 9 is the exponent or power. The exponent indicates how many times the base is multiplied by itself. Similarly, 3^5 means 3 multiplied by itself 5 times (3 × 3 × 3 × 3 × 3). Understanding this notation is crucial for manipulating exponential expressions effectively. When we encounter expressions involving exponents, we are essentially dealing with repeated multiplication. The rules of exponents provide shortcuts for simplifying these repeated multiplications, making calculations more manageable. In our example, we have two terms involving exponents, (2^9 × 3^5) and (2^4 × 3)^2, which we will simplify using these rules.

The Power of a Product Rule

One of the key rules we'll employ is the power of a product rule, which states that (ab)^n = a^n * b^n. This rule tells us that when a product raised to a power, we can distribute the power to each factor within the product. In our expression, we have (2^4 × 3)^2. Applying the power of a product rule, we get (2⁴⁾2 × 3^2. Now, we need to simplify (2⁴⁾2. This brings us to another important rule: the power of a power rule. Understanding and applying these rules correctly is essential for simplifying exponential expressions. Mistakes in applying these rules can lead to incorrect results. For example, incorrectly distributing the exponent or misunderstanding the order of operations can result in a wrong answer. Therefore, it's crucial to practice and internalize these rules to ensure accuracy.

The Power of a Power Rule

The power of a power rule states that (a^m)n = a^(mn). This means when we raise a power to another power, we multiply the exponents. So, (2⁴⁾2 becomes 2^(42) = 2^8. Now we have (2^4 × 3)^2 simplified to 2^8 × 3^2. This is a significant step in simplifying the original expression. We have successfully applied two important rules of exponents to break down a complex term into simpler components. By understanding and applying these rules systematically, we can simplify even more intricate expressions. The power of a power rule is particularly useful when dealing with nested exponents, where an exponent is raised to another exponent. Mastering this rule is essential for simplifying expressions in algebra and calculus.

Applying the Rules to the Expression

Now, let's apply these rules to our original expression: (2^9 × 3^5) × (2^4 × 3)^2. First, we simplify (2^4 × 3)^2 using the power of a product rule: (2⁴⁾2 × 3^2. Next, we apply the power of a power rule to (2⁴⁾2, which gives us 2^(4*2) = 2^8. So, (2^4 × 3)^2 simplifies to 2^8 × 3^2. Now we can rewrite the original expression as (2^9 × 3^5) × (2^8 × 3^2). We have successfully simplified one part of the expression, making it easier to combine with the other part. This step-by-step approach is crucial for tackling complex mathematical problems. Breaking down the problem into smaller, manageable parts makes the simplification process less daunting and reduces the likelihood of errors.

The Product of Powers Rule

Next, we use the product of powers rule, which states that a^m × a^n = a^(m+n). This rule tells us that when multiplying exponential expressions with the same base, we add the exponents. In our expression, we have 2^9 × 2^8 and 3^5 × 3^2. Applying the product of powers rule to the terms with base 2, we get 2^(9+8) = 2^17. Similarly, for the terms with base 3, we get 3^(5+2) = 3^7. This rule is fundamental to simplifying expressions involving exponents. It allows us to combine terms with the same base, making the expression more concise. The product of powers rule is a direct consequence of the definition of exponents as repeated multiplication. When we multiply a^m by a^n, we are essentially multiplying 'a' by itself 'm' times and then multiplying the result by 'a' multiplied by itself 'n' times. This is equivalent to multiplying 'a' by itself 'm+n' times, which is a^(m+n).

Final Simplification and the Answer

Combining these results, our simplified expression is 2^17 × 3^7. This matches option D. Therefore, the correct answer is D. 2^17 × 3^7. We have successfully simplified the original expression by applying the rules of exponents systematically. This demonstrates the power of these rules in simplifying complex mathematical expressions. The ability to simplify expressions is crucial in various areas of mathematics, including algebra, calculus, and trigonometry. It allows us to solve equations, analyze functions, and make predictions. Mastering the rules of exponents is an essential step in developing mathematical proficiency. By understanding and applying these rules correctly, we can simplify complex expressions and solve challenging problems with confidence.

Expressing the Answer in Different Forms

While 2^17 × 3^7 is the simplified form, it's worth noting that we could also calculate the numerical value, although it would be a very large number. Additionally, we can explore whether this expression can be written in other equivalent forms. For instance, we could try to express it as a power of 6, but since the exponents of 2 and 3 are different, this is not possible. Understanding the limitations and possibilities of different representations can deepen our understanding of the expression. Exploring different forms of an expression can sometimes reveal hidden properties or connections to other mathematical concepts. For example, in some cases, we might be able to factor the expression or rewrite it in a form that is easier to work with in a particular context. The ability to manipulate expressions in different ways is a valuable skill in mathematics.

Conclusion: Mastering Exponential Simplification

In conclusion, we have successfully simplified the expression (2^9 × 3^5) × (2^4 × 3)^2 to 2^17 × 3^7 by applying the power of a product rule, the power of a power rule, and the product of powers rule. This process highlights the importance of understanding and applying the rules of exponents correctly. Mastering these rules is crucial for simplifying exponential expressions and tackling more complex mathematical problems. By following a systematic approach and breaking down the problem into smaller steps, we can simplify even the most challenging expressions. Practice and familiarity with these rules will build your confidence and proficiency in mathematics. Remember, the key to success in mathematics is a solid understanding of the fundamental principles and consistent practice. With dedication and effort, you can master exponential simplification and unlock new levels of mathematical understanding.