Calculating Exponential Expressions A Step By Step Guide

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Understanding the Problem: Simplifying Exponential Expressions

In this mathematical problem, we are tasked with calculating the value of a fractional expression involving exponents. The expression, $ rac{\left(3^4 \cdot 33\right)2}{9^3}$, requires us to simplify both the numerator and the denominator using the rules of exponents before performing the division. This involves understanding how to multiply exponents with the same base, how to raise a power to a power, and how to express numbers as powers of a common base. Successfully navigating these steps will lead us to the simplified numerical answer. Let's break down the process step-by-step to ensure a clear understanding of the solution. We will start by simplifying the numerator, then the denominator, and finally, we will perform the division to arrive at the final result. This approach will not only give us the answer but also reinforce the fundamental principles of exponential arithmetic.

Exponents are a shorthand way of representing repeated multiplication. For instance, 343^4 means 3 multiplied by itself four times (3 * 3 * 3 * 3). When we encounter expressions with exponents, there are several rules we can apply to simplify them. One of the most fundamental rules is the product of powers rule, which states that when multiplying two exponential terms with the same base, we add their exponents. Mathematically, this is represented as amâ‹…an=am+na^m \cdot a^n = a^{m+n}. This rule is crucial for simplifying the numerator of our expression. Another important rule is the power of a power rule, which states that when raising a power to another power, we multiply the exponents. This is represented as (am)n=amâ‹…n(a^m)^n = a^{m \cdot n}. This rule will be applied after we simplify the expression within the parentheses in the numerator. Finally, we need to understand how to express numbers as powers of a common base. In our case, we have 9 in the denominator, which can be expressed as 323^2. This transformation allows us to combine the numerator and the denominator into a single exponential term, making the final calculation straightforward. By mastering these exponent rules, we can tackle complex expressions with confidence and accuracy. The beauty of mathematics lies in its ability to simplify complex problems into manageable steps, and this problem is a perfect example of that principle.

To successfully solve this problem, a systematic approach is essential. First, we will focus on simplifying the numerator, which involves applying the product of powers rule to the expression inside the parentheses and then applying the power of a power rule. Second, we will simplify the denominator by expressing 9 as a power of 3. This step is crucial for unifying the bases in the numerator and the denominator, allowing us to combine the terms more easily. Finally, we will divide the simplified numerator by the simplified denominator. This division will involve subtracting the exponents, another fundamental rule of exponents. This step-by-step method not only helps in solving the problem accurately but also reinforces the understanding of the underlying principles of exponents. By following this structured approach, we minimize the chances of error and gain a deeper appreciation for the elegance of mathematical problem-solving. Each step builds upon the previous one, leading us logically to the final solution. The process is as important as the answer itself, as it enhances our mathematical reasoning and problem-solving skills.

Step-by-Step Solution

Let's begin by focusing on the numerator: (34â‹…33)2\left(3^4 \cdot 3^3\right)^2. The first step is to simplify the expression inside the parentheses. According to the product of powers rule, when multiplying exponential terms with the same base, we add the exponents. Therefore, 34â‹…33=34+3=373^4 \cdot 3^3 = 3^{4+3} = 3^7. Now, we have (37)2\left(3^7\right)^2. To further simplify this, we apply the power of a power rule, which states that when raising a power to another power, we multiply the exponents. Thus, (37)2=37â‹…2=314\left(3^7\right)^2 = 3^{7 \cdot 2} = 3^{14}. The numerator is now simplified to 3143^{14}.

Next, we turn our attention to the denominator: 939^3. To simplify this, we need to express 9 as a power of 3. We know that 9=329 = 3^2, so we can rewrite the denominator as (32)3(3^2)^3. Again, we apply the power of a power rule, which gives us (32)3=32â‹…3=36(3^2)^3 = 3^{2 \cdot 3} = 3^6. The denominator is now simplified to 363^6.

Now that we have simplified both the numerator and the denominator, we can rewrite the original expression as $ rac{3{14}}{36}$. To perform this division, we use the quotient of powers rule, which states that when dividing exponential terms with the same base, we subtract the exponents. Therefore, $\frac{3{14}}{36} = 3^{14-6} = 3^8$. To find the numerical value of 383^8, we multiply 3 by itself eight times: 38=3â‹…3â‹…3â‹…3â‹…3â‹…3â‹…3â‹…3=65613^8 = 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 = 6561.

Therefore, the final answer to the expression $\frac{\left(3^4 \cdot 33\right)2}{9^3}$ is 6561. This solution demonstrates the power and efficiency of using the rules of exponents to simplify complex expressions. By breaking down the problem into smaller, manageable steps, we were able to arrive at the solution with clarity and precision. The process not only provides the answer but also reinforces our understanding of the fundamental principles of exponential arithmetic. This type of problem is a classic example of how mathematical rules can be applied to solve seemingly complex calculations, highlighting the elegance and power of mathematical thinking.

Key Exponent Rules Used

Throughout the solution, we utilized several key exponent rules. Understanding these rules is fundamental to solving problems involving exponents. Let's recap the rules that were crucial in simplifying the given expression.

  1. Product of Powers Rule: This rule states that when multiplying exponential terms with the same base, we add the exponents. Mathematically, it is represented as amâ‹…an=am+na^m \cdot a^n = a^{m+n}. This rule was used to simplify the expression inside the parentheses in the numerator. By adding the exponents of the terms with the same base (3), we were able to combine them into a single term. This is a cornerstone of exponential arithmetic and is frequently used in simplifying expressions. The ability to quickly apply this rule is essential for efficiently solving problems involving exponents.

  2. Power of a Power Rule: This rule states that when raising a power to another power, we multiply the exponents. Mathematically, it is represented as (am)n=amâ‹…n(a^m)^n = a^{m \cdot n}. We applied this rule twice: first, to simplify the numerator after applying the product of powers rule, and second, to simplify the denominator after expressing 9 as a power of 3. This rule is particularly useful when dealing with nested exponents, as it allows us to reduce the expression to a single exponent. Mastering this rule is crucial for simplifying complex exponential expressions.

  3. Quotient of Powers Rule: This rule states that when dividing exponential terms with the same base, we subtract the exponents. Mathematically, it is represented as $\frac{am}{an} = a^{m-n}$. This rule was used to simplify the fraction after we had simplified the numerator and the denominator. By subtracting the exponents, we were able to reduce the fraction to a single exponential term. This rule is the inverse operation of the product of powers rule and is equally important in simplifying exponential expressions.

By understanding and applying these three rules, we were able to systematically simplify the given expression and arrive at the final answer. These rules form the foundation of exponential arithmetic and are essential for solving a wide range of mathematical problems. Practice in applying these rules will build confidence and fluency in working with exponents. The ability to recognize when and how to apply these rules is a key skill in mathematics, enabling us to tackle complex problems with ease and efficiency. These rules are not just abstract concepts; they are powerful tools that can simplify calculations and reveal the underlying structure of mathematical expressions.

Conclusion: The Power of Exponential Rules

In conclusion, by applying the fundamental rules of exponents – the product of powers rule, the power of a power rule, and the quotient of powers rule – we successfully calculated the value of the expression $\frac{\left(3^4 \cdot 33\right)2}{9^3}$. We systematically simplified the numerator and the denominator, and then performed the division to arrive at the final answer of 6561. This problem highlights the importance of understanding and applying these exponent rules to efficiently solve mathematical problems.

The process of simplifying exponential expressions not only provides a numerical answer but also reinforces the principles of mathematical reasoning and problem-solving. By breaking down a complex problem into smaller, manageable steps, we can approach it with clarity and confidence. Each step in the solution builds upon the previous one, leading us logically to the final result. This systematic approach is a valuable skill that can be applied to various mathematical and real-world problems. The ability to manipulate exponents and simplify expressions is a fundamental skill in algebra and is crucial for further studies in mathematics and related fields.

The power of exponential rules lies in their ability to transform complex calculations into simpler ones. These rules are not just abstract concepts; they are powerful tools that can simplify calculations and reveal the underlying structure of mathematical expressions. By mastering these rules, we gain a deeper understanding of mathematical principles and enhance our ability to solve problems efficiently. The elegance of mathematics is often found in its ability to simplify complex ideas into manageable components, and this problem is a perfect example of that principle. As we continue to explore mathematics, the principles learned in this problem will serve as a foundation for more advanced concepts and techniques. The journey of mathematical learning is a continuous process of building upon fundamental principles, and the mastery of exponent rules is a crucial step in that journey.