Simplifying Expressions With Fractional Exponents A Step-by-Step Guide

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In the realm of mathematics, expressions involving fractional exponents often pose a challenge. However, with a systematic approach and a solid understanding of exponent rules, these expressions can be simplified effectively. This comprehensive guide delves into the simplification of a complex expression with fractional exponents, providing a step-by-step solution and insightful explanations. Let's embark on this mathematical journey and unravel the intricacies of fractional exponents.

Understanding the Fundamentals of Fractional Exponents

Fractional exponents represent a powerful way to express both roots and powers simultaneously. A fractional exponent, such as a^(m/n), signifies the nth root of a raised to the power of m. In simpler terms, it can be interpreted as (n√a)^m or n√(a^m). The denominator of the fraction (n) indicates the root to be taken, while the numerator (m) represents the power to which the base is raised. Grasping this fundamental concept is crucial for navigating the intricacies of expressions with fractional exponents. When working with fractional exponents, several key rules come into play, and mastering these rules is essential for simplifying expressions effectively. One of the most fundamental rules is the product of powers rule, which states that when multiplying exponents with the same base, you add the powers: x^a * x^b = x^(a+b). This rule allows us to combine terms with the same base, simplifying the expression. Similarly, the quotient of powers rule dictates that when dividing exponents with the same base, you subtract the powers: x^a / x^b = x^(a-b). This rule is invaluable for simplifying fractions involving exponents. Another critical rule is the power of a power rule, which states that when raising a power to another power, you multiply the exponents: (xa)b = x^(ab)*. This rule simplifies expressions where exponents are nested within each other. Furthermore, understanding negative exponents is crucial. A negative exponent indicates the reciprocal of the base raised to the positive value of the exponent: x^(-a) = 1/x^a. This rule allows us to move terms with negative exponents from the numerator to the denominator or vice versa, simplifying the expression. By mastering these fundamental rules, we equip ourselves with the tools necessary to tackle complex expressions involving fractional exponents.

Breaking Down the Expression: A Step-by-Step Approach

Let's consider the expression:

(x^(2/7) / z^(-1/2)) * (x^(2/5) / z^(2/3)) * (x^(-9/7) / z^(2/3)) * (z^(5/6) / x^(-3/5))

To simplify this expression, we will systematically apply the exponent rules discussed earlier. Our primary goal is to combine terms with the same base (x and z) by adding or subtracting their exponents. We'll break down the simplification process into manageable steps, ensuring clarity and accuracy.

Step 1: Rearranging the Terms

First, let's rearrange the expression to group terms with the same base together. This will make it easier to apply the exponent rules. We can rewrite the expression as:

(x^(2/7) * x^(2/5) * x^(-9/7) * x^(-3/5)) * (z^(-1/2) / z^(2/3) / z^(2/3) * z^(5/6))

By grouping the x terms and the z terms, we create a clearer structure for simplification. This rearrangement allows us to focus on combining exponents with the same base, making the process more organized and less prone to errors. Rearranging terms is a crucial first step in simplifying complex expressions, as it sets the stage for applying exponent rules effectively.

Step 2: Combining x Terms

Now, let's focus on the x terms. We'll apply the product of powers rule, which states that when multiplying exponents with the same base, you add the powers. So, we have:

x^(2/7 + 2/5 + (-9/7) + (-3/5))

To add these fractions, we need to find a common denominator. The least common multiple of 7 and 5 is 35. So, we convert the fractions:

x^((10/35) + (14/35) + (-45/35) + (-21/35))

Now, we can add the numerators:

x^((10 + 14 - 45 - 21) / 35)

x^(-42/35)

We can simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is 7:

x^(-6/5)

Therefore, the combined x terms simplify to x^(-6/5). This result demonstrates the power of the product of powers rule in simplifying expressions with fractional exponents. By finding a common denominator and adding the exponents, we efficiently combined multiple x terms into a single term.

Step 3: Combining z Terms

Next, let's simplify the z terms. We have:

z^(-1/2) / (z^(2/3) * z^(2/3)) * z^(5/6)

First, we apply the product of powers rule to the denominator:

z^(-1/2) / z^(2/3 + 2/3) * z^(5/6)

z^(-1/2) / z^(4/3) * z^(5/6)

Now, we apply the quotient of powers rule, which states that when dividing exponents with the same base, you subtract the powers:

z^((-1/2) - (4/3) + (5/6))

To subtract and add these fractions, we need a common denominator. The least common multiple of 2, 3, and 6 is 6. So, we convert the fractions:

z^((-3/6) - (8/6) + (5/6))

Now, we can add and subtract the numerators:

z^((-3 - 8 + 5) / 6)

z^(-6/6)

z^(-1)

Therefore, the combined z terms simplify to z^(-1). This simplification showcases the combined application of the product of powers and quotient of powers rules. By systematically adding and subtracting exponents with a common base, we efficiently reduced the multiple z terms to a single term.

Step 4: Final Simplification

Now that we have simplified the x and z terms separately, let's combine them to get the final simplified expression:

x^(-6/5) * z^(-1)

To eliminate negative exponents, we can rewrite the expression as:

1 / (x^(6/5) * z)

This is the simplified form of the original expression. We have successfully navigated the complexities of fractional exponents by applying the rules systematically and breaking down the problem into manageable steps. The final expression, 1 / (x^(6/5) * z), represents the most concise and simplified form of the original expression.

Conclusion: Mastering Fractional Exponents

Simplifying expressions with fractional exponents may seem daunting at first, but by understanding the fundamental rules and applying a systematic approach, these expressions can be tamed. This comprehensive guide has demonstrated a step-by-step method for simplifying a complex expression, highlighting the importance of exponent rules and careful manipulation. With practice and a solid grasp of the concepts, you too can master the art of simplifying expressions with fractional exponents. Remember, the key to success lies in understanding the underlying principles and applying them consistently. As you encounter more complex expressions, the skills you've developed here will serve as a solid foundation for further mathematical exploration.

The Value of (x^(2/7) / z^(-1/2)) * (x^(2/5) / z^(2/3)) * (x^(-9/7) / z^(2/3)) * (z^(5/6) / x^(-3/5)): A Detailed Solution

Question

Determine the value of the expression:

(x^(2/7) / z^(-1/2)) * (x^(2/5) / z^(2/3)) * (x^(-9/7) / z^(2/3)) * (z^(5/6) / x^(-3/5))

(a) 1 (b) -1 (c) 0

Detailed Solution

To find the value of the given expression, we need to simplify it by applying the rules of exponents. Here’s a step-by-step solution:

  1. Rewrite the Expression

First, let’s rewrite the expression to group the terms with the same base (x and z) together:

(x^(2/7) / z^(-1/2)) * (x^(2/5) / z^(2/3)) * (x^(-9/7) / z^(2/3)) * (z^(5/6) / x^(-3/5))

This can be rearranged as:

(x^(2/7) * x^(2/5) * x^(-9/7) * x^(3/5)) * (z^(1/2) * z^(-2/3) * z^(-2/3) * z^(5/6))

  1. Combine the x Terms

Combine the x terms by adding their exponents:

x^(2/7 + 2/5 - 9/7 + 3/5)

To add these fractions, we need a common denominator, which is 35:

x^((10/35) + (14/35) - (45/35) + (21/35))

Now, add the numerators:

x^((10 + 14 - 45 + 21) / 35)

x^(0/35)

x^0

Any non-zero number raised to the power of 0 is 1:

x^0 = 1

  1. Combine the z Terms

Combine the z terms by adding their exponents:

z^(1/2 - 2/3 - 2/3 + 5/6)

To add these fractions, we need a common denominator, which is 6:

z^((3/6) - (4/6) - (4/6) + (5/6))

Now, add the numerators:

z^((3 - 4 - 4 + 5) / 6)

z^(0/6)

z^0

Any non-zero number raised to the power of 0 is 1:

z^0 = 1

  1. Final Simplification

Now, multiply the simplified x and z terms:

1 * 1 = 1

Thus, the value of the expression is 1.

Answer

(a) 1

Conclusion

This problem demonstrates the importance of understanding and applying the rules of exponents. By systematically grouping like terms, finding common denominators, and simplifying, we can efficiently solve complex expressions. In this case, the final answer is 1, showcasing the elegance of mathematical simplification.