Simplifying Expressions With Rational Exponents
Understanding and simplifying expressions with rational exponents is a fundamental skill in algebra. This article will guide you through the process of simplifying the expression (8x³y⁻¹ / x³y⁻²)⁻¹/³, assuming all variables are positive. We will break down the steps, explain the rules of exponents, and provide a clear, step-by-step solution. Mastering this skill will not only help you in your math courses but also in various fields that require mathematical applications. Rational exponents can seem tricky at first, but with a solid understanding of the underlying principles, you can confidently tackle even the most complex expressions. The key is to remember the rules of exponents and apply them systematically. By the end of this article, you'll have a clear understanding of how to simplify expressions like this one, and you'll be able to apply these techniques to other similar problems. So, let's dive in and unlock the secrets of simplifying expressions with rational exponents.
Understanding Rational Exponents
Before we dive into the simplification, it's crucial to understand what rational exponents are and how they work. A rational exponent is an exponent that can be expressed as a fraction, where the numerator indicates the power to which the base is raised, and the denominator indicates the root to be taken. For example, x^(m/n) can be rewritten as the nth root of x raised to the power of m, or (√n)^m. This understanding is the cornerstone of simplifying expressions with rational exponents. When dealing with expressions involving rational exponents, it's essential to remember the basic rules of exponents. These rules provide the foundation for simplifying complex expressions into more manageable forms. For instance, when you multiply expressions with the same base, you add the exponents (x^a * x^b = x^(a+b)). Similarly, when you divide expressions with the same base, you subtract the exponents (x^a / x^b = x^(a-b)). Understanding these rules is critical for successfully simplifying expressions. Another important rule to remember is the power of a power rule, which states that when you raise a power to another power, you multiply the exponents ((xa)b = x^(a*b)). This rule is particularly useful when dealing with expressions like the one we're going to simplify. Additionally, a negative exponent indicates the reciprocal of the base raised to the positive exponent (x^-a = 1/x^a). This rule helps us deal with negative exponents effectively. Finally, any number raised to the power of 0 is 1 (x^0 = 1), which can sometimes simplify expressions significantly.
Step-by-Step Simplification Process
Let's break down the given expression (8x³y⁻¹ / x³y⁻²)⁻¹/³ step by step. Our goal is to simplify this expression using the rules of rational exponents, assuming all variables are positive. The first step in simplifying this expression is to address the terms inside the parentheses. We have a fraction with terms involving x and y, and we can use the quotient rule for exponents to simplify these terms. The quotient rule states that when dividing terms with the same base, we subtract the exponents. So, for the x terms, we have x³ / x³, which simplifies to x^(3-3) = x^0 = 1. For the y terms, we have y⁻¹ / y⁻², which simplifies to y^(-1 - (-2)) = y^(-1 + 2) = y^1 = y. After simplifying the x and y terms, the expression inside the parentheses becomes much simpler. Now, let's address the constant term. We have 8 in the numerator, and since all the x terms have canceled out, we are left with 8y inside the parentheses. This simplified form makes the next steps much easier to manage. The next step is to deal with the exponent outside the parentheses, which is -1/3. This exponent applies to the entire expression inside the parentheses, including both the constant term and the variable. Remember that a negative exponent means we need to take the reciprocal of the base, and the fractional exponent indicates a root. So, we will apply this exponent to both 8 and y separately. Applying the exponent -1/3 to 8, we get 8^(-1/3). This can be rewritten as 1 / (8^(1/3)). The cube root of 8 is 2, so 8^(1/3) = 2. Therefore, 8^(-1/3) = 1/2. Next, we apply the exponent -1/3 to y, which gives us y^(-1/3). This can be rewritten as 1 / y^(1/3). So, now we have simplified the expression to a point where we can combine the simplified terms. Let's put everything together and see what our final simplified expression looks like.
Applying the Rules of Exponents
To begin, let's rewrite the expression: (8x³y⁻¹ / x³y⁻²)⁻¹/³. The first step involves simplifying the fraction inside the parentheses. We can use the quotient rule of exponents, which states that a^m / a^n = a^(m-n). Applying this rule to the x terms, we have x³ / x³ = x^(3-3) = x^0 = 1. This simplifies the x terms to 1, effectively eliminating them from the numerator and denominator. Next, let's apply the quotient rule to the y terms. We have y⁻¹ / y⁻² = y^(-1 - (-2)) = y^(-1 + 2) = y^1 = y. So, the y terms simplify to y. Now, let's consider the constant term, which is 8 in the numerator. After simplifying the x and y terms, the expression inside the parentheses becomes 8y. The expression now looks much simpler: (8y)⁻¹/³. The next step is to apply the exponent -1/3 to the entire expression inside the parentheses. Remember that a negative exponent means we take the reciprocal, and a fractional exponent indicates a root. So, we need to apply this exponent to both 8 and y separately. Applying the exponent -1/3 to 8, we get 8^(-1/3). This is equivalent to 1 / (8^(1/3)). The cube root of 8 is 2, so 8^(1/3) = 2. Therefore, 8^(-1/3) = 1/2. Next, we apply the exponent -1/3 to y, which gives us y^(-1/3). This can be rewritten as 1 / y^(1/3). Now we have simplified the expression to a point where we can combine the simplified terms. We have 1/2 from simplifying 8^(-1/3) and 1 / y^(1/3) from simplifying y^(-1/3).
Detailed Solution
Now, let's combine the simplified terms. We have (1/2) * (1 / y^(1/3)). This simplifies to 1 / (2y^(1/3)). This is a simplified form of the expression, but we can take it a step further to rationalize the denominator. To rationalize the denominator, we need to eliminate the radical (the fractional exponent) from the denominator. In this case, we have y^(1/3) in the denominator, which represents the cube root of y. To eliminate this, we need to multiply both the numerator and the denominator by a term that will make the exponent of y in the denominator a whole number. Since we have y^(1/3), we need to multiply by y^(2/3) to get y^(1/3) * y^(2/3) = y^(1/3 + 2/3) = y^1 = y. So, we multiply both the numerator and the denominator by y^(2/3). This gives us (1 * y^(2/3)) / (2y^(1/3) * y^(2/3)) = y^(2/3) / (2y). Now, the expression is simplified and the denominator is rationalized. Our final simplified expression is y^(2/3) / (2y). This is the most simplified form of the original expression (8x³y⁻¹ / x³y⁻²)⁻¹/³, assuming all variables are positive. In summary, we started with a complex expression involving rational exponents and used the rules of exponents to simplify it step by step. We first simplified the terms inside the parentheses, then applied the exponent outside the parentheses, and finally rationalized the denominator to arrive at the most simplified form. This process highlights the importance of understanding and applying the rules of exponents effectively. By breaking down the problem into smaller steps, we can make even complex expressions manageable and arrive at the correct solution. This methodical approach is crucial for success in algebra and other areas of mathematics. The ability to simplify expressions with rational exponents is a valuable skill that can be applied in various contexts, making it an essential part of mathematical literacy.
Final Simplified Expression
After following all the steps, the simplified expression is:
y^(2/3) / (2y)
This result is obtained by applying the rules of exponents, simplifying within the parentheses, handling the negative exponent, and rationalizing the denominator. Remember, the key to simplifying these types of expressions is a solid understanding of exponent rules and a methodical approach.
Conclusion
Simplifying expressions with rational exponents involves a systematic application of exponent rules and a clear understanding of how fractional exponents work. By breaking down the problem into smaller, manageable steps, we can tackle even complex expressions with confidence. Remember to simplify within parentheses first, apply the power rule, handle negative exponents, and rationalize the denominator if necessary. This methodical approach will help you navigate these problems effectively and arrive at the correct solution. Mastering these techniques not only enhances your algebraic skills but also provides a strong foundation for more advanced mathematical concepts. Practice is key to developing proficiency in simplifying expressions with rational exponents. Work through various examples, and don't hesitate to review the rules and steps as needed. With consistent effort, you'll become more comfortable and confident in your ability to simplify these expressions. Remember, mathematics is a journey of continuous learning and growth. Embrace the challenges, and celebrate your successes along the way. The skills you develop in algebra will serve you well in many areas of life, both academic and professional. Keep exploring, keep learning, and keep simplifying!
For further information on rational exponents and related topics, you can visit Khan Academy's exponents and radicals section.
