Simplifying Exponential Expressions A Step By Step Guide

JU07/15/2025 08, 2025 by THE IDEN 57 views

Simplifying complex mathematical expressions can seem daunting, but by breaking them down into smaller, manageable parts, we can arrive at the solution systematically. This article will guide you through the process of simplifying a given expression involving exponents and fractions. Let's dive into the problem and explore the steps to simplify it effectively. This article aims to provide a comprehensive understanding of how to simplify expressions involving fractional exponents and different bases. We'll start by identifying the key components of the expression and then proceed step-by-step to break down each term. By the end of this guide, you'll have a clear understanding of the methods and techniques used to simplify such expressions, making it easier for you to tackle similar problems in the future. This article emphasizes the importance of understanding the properties of exponents and how to apply them effectively in simplifying expressions. We will also highlight common mistakes and how to avoid them, ensuring that you develop a solid foundation in this area of mathematics. So, let’s embark on this journey to simplify complex expressions and enhance your mathematical skills.

The Expression

We are given the following expression to simplify:

${\frac{81^{\frac{3}{2}} \times 27^{\frac{4}{3}} \times 64^{\frac{3}{4}}}{2^{\frac{7}{2}} \times 3^{\frac{1}{2}} \times 4^{\frac{1}{2}} \times 8^{\frac{7}{2}}}}}$

This expression involves various numbers raised to fractional powers. To simplify it, we will break it down step by step, using the properties of exponents and prime factorization.

Step 1: Prime Factorization

The first step in simplifying this expression is to express each base number as a product of its prime factors. This will allow us to apply the properties of exponents more easily. Let's break down each base:

81 can be written as 3^4
27 can be written as 3^3
64 can be written as 2^6
4 can be written as 2^2
8 can be written as 2^3

Now, we substitute these prime factorizations back into the original expression. Prime factorization is a crucial step in simplifying expressions with exponents, especially when dealing with fractional powers. It allows us to rewrite the bases in terms of their prime factors, which then makes it easier to apply the exponent rules. For instance, breaking down 81 into 3^4 allows us to use the power of a power rule, simplifying the expression significantly. Understanding prime factorization is not just beneficial for this type of problem but also for various other mathematical concepts such as finding the greatest common divisor (GCD) and the least common multiple (LCM). By mastering this technique, you lay a strong foundation for more advanced algebraic manipulations. Moreover, prime factorization helps in identifying common factors between the numerator and the denominator, which is essential for simplifying fractions. This method ensures that we are working with the smallest possible bases, making the subsequent calculations more manageable and less prone to errors. In summary, prime factorization is a fundamental tool in simplifying expressions, and its importance cannot be overstated. This initial step sets the stage for further simplification and ensures accuracy in the final result.

Step 2: Substitute Prime Factors

Substituting the prime factors into the expression, we get:

${\frac{(3^4)^{\frac{3}{2}} \times (3^3)^{\frac{4}{3}} \times (2^6)^{\frac{3}{4}}}{2^{\frac{7}{2}} \times 3^{\frac{1}{2}} \times (2^2)^{\frac{1}{2}} \times (2^3)^{\frac{7}{2}}}}}$

This substitution allows us to work with the powers of prime numbers, which simplifies the application of exponent rules. Substituting the prime factors is a critical step because it transforms the original expression into a form where exponent rules can be readily applied. By expressing each base as a power of its prime factors, we can then use rules such as the power of a power rule, which states that (a^m)n = a^(m*n). This step ensures that we are working with the simplest possible bases, making subsequent calculations more straightforward and accurate. For instance, substituting 81 with 3^4 and 27 with 3^3 allows us to combine and simplify the terms involving the base 3 more effectively. This approach is not only efficient but also minimizes the chances of making errors. Additionally, this step prepares the expression for further simplification by allowing us to combine like terms and cancel out common factors in the numerator and denominator. In essence, substituting prime factors is a strategic move that sets the stage for a clear and concise simplification process. It helps to reduce the complexity of the expression and ensures that we are working with the most fundamental components.

Step 3: Apply the Power of a Power Rule

Now, we apply the power of a power rule, which states that ${(a^m)^n = a^{m \times n}}$ to each term in the expression:

${\frac{3^{4 \times \frac{3}{2}} \times 3^{3 \times \frac{4}{3}} \times 2^{6 \times \frac{3}{4}}}{2^{\frac{7}{2}} \times 3^{\frac{1}{2}} \times 2^{2 \times \frac{1}{2}} \times 2^{3 \times \frac{7}{2}}}}}$

This rule helps us simplify the exponents by multiplying them. Applying the power of a power rule is a fundamental technique in simplifying expressions with exponents. This rule, which states that (a^m)n = a^(mn), allows us to multiply the exponents when a power is raised to another power. In our case, applying this rule simplifies terms like (3⁴⁾(3/2) to 3^(4(3/2)), which then becomes 3^6. This step is crucial because it reduces the complexity of the exponents and prepares the expression for further simplification. By applying this rule systematically, we can eliminate nested exponents and work with single exponents, making the expression more manageable. Furthermore, this rule is universally applicable across different bases and fractional exponents, making it a versatile tool in simplifying various mathematical expressions. It is essential to apply this rule correctly to ensure accuracy in the final result. In summary, the power of a power rule is a cornerstone of exponent manipulation, enabling us to simplify complex expressions efficiently and accurately.

Step 4: Simplify the Exponents

Next, we simplify the exponents by performing the multiplications:

${\frac{3^6 \times 3^4 \times 2^{\frac{9}{2}}}{2^{\frac{7}{2}} \times 3^{\frac{1}{2}} \times 2^1 \times 2^{\frac{21}{2}}}}}$

This step involves simple arithmetic to find the new exponents. Simplifying the exponents is a crucial step in reducing the complexity of the expression. This involves performing the multiplication of the exponents as dictated by the power of a power rule. For example, in our expression, we simplify 4 * (3/2) to 6, 3 * (4/3) to 4, and 6 * (3/4) to 9/2. These calculations result in a more straightforward representation of the exponents, making the subsequent steps easier to manage. Simplifying the exponents not only makes the expression more readable but also prepares it for further operations such as combining like terms and canceling out common factors. This step requires careful arithmetic to avoid errors, as any mistake in the exponent calculation can propagate through the rest of the solution. By ensuring accurate simplification of the exponents, we maintain the integrity of the expression and pave the way for a correct final answer. In essence, this step is a critical bridge in the simplification process, transforming complex exponents into simpler, manageable forms.

Step 5: Combine Like Terms

Now, we combine the terms with the same base by adding their exponents in the numerator and the denominator:

${\frac{3^{6+4} \times 2^{\frac{9}{2}}}{2^{\frac{7}{2}+1+\frac{21}{2}} \times 3^{\frac{1}{2}}}}}$

${\frac{3^{10} \times 2^{\frac{9}{2}}}{2^{\frac{7+2+21}{2}} \times 3^{\frac{1}{2}}}}}$

${\frac{3^{10} \times 2^{\frac{9}{2}}}{2^{\frac{30}{2}} \times 3^{\frac{1}{2}}}}}$

${\frac{3^{10} \times 2^{\frac{9}{2}}}{2^{15} \times 3^{\frac{1}{2}}}}}$

Combining like terms is a pivotal step in simplifying expressions, as it consolidates terms with the same base by applying the rule a^m * a^n = a^(m+n). This process reduces the number of terms and simplifies the overall expression. In our case, we combine the powers of 3 and the powers of 2 separately in both the numerator and the denominator. For the numerator, we add the exponents of 3 (6 and 4) to get 3^10, and for the denominator, we add the exponents of 2 (7/2, 1, and 21/2) to get 2^15. This step not only simplifies the expression but also prepares it for the next stage, where we will divide terms with the same base. By combining like terms, we reduce redundancy and make the expression more manageable, which is essential for achieving the final simplified form. Furthermore, this step highlights the importance of understanding and applying exponent rules correctly. In summary, combining like terms is a critical technique that streamlines the simplification process and sets the stage for the final solution.

Step 6: Divide Terms with the Same Base

Next, we divide the terms with the same base by subtracting their exponents:

${3^{10 - \frac{1}{2}} \times 2^{\frac{9}{2} - 15}}$

${3^{\frac{20-1}{2}} \times 2^{\frac{9-30}{2}}}$

${3^{\frac{19}{2}} \times 2^{-\frac{21}{2}}}$

Dividing terms with the same base is another crucial step in simplifying expressions, relying on the rule a^m / a^n = a^(m-n). This process involves subtracting the exponent in the denominator from the exponent in the numerator for terms with the same base. In our expression, we divide the powers of 3 and the powers of 2 separately. For the base 3, we subtract 1/2 from 10, resulting in 3^(19/2), and for the base 2, we subtract 15 from 9/2, resulting in 2^(-21/2). This step significantly reduces the complexity of the expression by eliminating the fractional format and combining terms with the same base. It also helps to isolate the remaining exponents, making it easier to express the final simplified form. The accuracy of this step is vital, as any error in subtracting the exponents can lead to an incorrect final answer. In summary, dividing terms with the same base is a key technique that streamlines the simplification process, allowing us to express the final result in a concise and manageable format.

Step 7: Simplify Negative Exponent

To eliminate the negative exponent, we move the term with the negative exponent to the denominator:

${\frac{3^{\frac{19}{2}}}{2^{\frac{21}{2}}}}}$

Simplifying negative exponents is an essential step in presenting the final simplified form of an expression. A negative exponent indicates that the base should be moved to the opposite side of the fraction (numerator to denominator or vice versa). The rule for this operation is a^(-n) = 1/a^n. In our expression, we have 2^(-21/2), which means we need to move 2^(21/2) from the numerator to the denominator. This transformation eliminates the negative exponent, making the expression more conventional and easier to interpret. This step is crucial for adhering to the standard mathematical practice of avoiding negative exponents in the final answer. Furthermore, simplifying negative exponents often helps in better understanding the magnitude of the terms involved. In summary, simplifying negative exponents is a key technique for presenting a mathematical expression in its most simplified and easily understandable form.

Step 8: Final Simplified Expression

The final simplified expression is:

${\frac{3^{\frac{19}{2}}}{2^{\frac{21}{2}}}}}$

This is the simplified form of the original expression, with all exponents and bases reduced to their simplest forms. The final simplified expression is the culmination of all the steps we have taken, presenting the result in its most concise and manageable form. In our case, the simplified expression is ${\frac{3^{\frac{19}{2}}}{2^{\frac{21}{2}}}}$ , which represents the original complex expression in a much cleaner and more understandable format. Achieving this final form requires a thorough understanding of exponent rules, prime factorization, and careful execution of each step. The simplified expression not only provides the answer but also showcases the mathematical elegance of reducing a complex problem into its fundamental components. This final step underscores the importance of accuracy and precision throughout the simplification process. In summary, the final simplified expression is the ultimate goal, demonstrating the power of mathematical techniques in solving complex problems.

Conclusion

By following these steps, we have successfully simplified the given expression. Breaking down the problem into smaller parts and applying the properties of exponents systematically allowed us to arrive at the solution. Remember, practice is key to mastering these techniques. In conclusion, simplifying complex expressions requires a systematic approach and a solid understanding of exponent rules. By breaking down the problem into smaller, manageable steps, we can simplify even the most daunting expressions. Our journey through this specific expression involved prime factorization, applying the power of a power rule, combining like terms, dividing terms with the same base, and simplifying negative exponents. Each step is crucial in the process, and accuracy at each stage is essential to reach the final simplified form. Practice is the key to mastering these techniques, and the more you practice, the more proficient you will become in simplifying complex mathematical expressions. This skill is not only valuable in academic settings but also in various real-world applications where mathematical simplification is necessary. So, keep practicing and honing your skills, and you will find that even the most complex expressions can be simplified with the right approach.