Simplifying Expressions With Fractional Exponents

In the realm of mathematics, simplifying expressions is a fundamental skill. This article dives into the process of simplifying an expression involving fractional exponents and negative exponents. Understanding these concepts is crucial for success in algebra and beyond. Let's break down the expression $\frac{5 a^{1 / 3} b^{-2}}{\left(10 a^{-2 / 3} b^{1 / 4}\right)^2}$ step-by-step.

Understanding the Basics of Exponents

Before we tackle the main expression, let's recap the rules of exponents. These rules form the bedrock of our simplification process. Exponents indicate how many times a base is multiplied by itself. For example, $x^3$ means $x * x * x$. When we deal with fractional exponents, we're essentially talking about roots and powers. A fractional exponent like $a^{m/n}$ can be interpreted as the nth root of a raised to the mth power, or $(\sqrt[n]{a})^m$. Negative exponents, on the other hand, indicate reciprocals. So, $b^{-2}$ is the same as $\frac{1}{b^2}$. Grasping these basics is paramount to simplifying complex expressions effectively. These rules aren't just abstract concepts; they're tools that allow us to manipulate and simplify mathematical statements, making them easier to understand and work with. When simplifying expressions, remember that each term and exponent plays a specific role. Identifying these roles is the key to successfully applying the rules of exponents. In the following sections, we’ll demonstrate how these rules come into play as we simplify our target expression.

Breaking Down the Expression

Let’s begin simplifying the expression $\frac{5 a^{1 / 3} b^{-2}}{\left(10 a^{-2 / 3} b^{1 / 4}\right)^2}$. The first step is to address the denominator, specifically the term raised to the power of 2. According to the power of a product rule, $(xy)^n = x^n y^n$, we need to apply the exponent outside the parenthesis to each term inside. This means we square both the coefficient and the variables with their respective exponents. Hence, $\left(10 a^{-2 / 3} b^{1 / 4}\right)^2$ becomes $10^2 * (a^{-2 / 3})^2 * (b^{1 / 4})^2$. When raising a power to a power, we multiply the exponents: $(x^m)^n = x^{mn}$. Applying this rule, $(a^{-2 / 3})^2$ becomes $a^{(-2 / 3) * 2} = a^{-4 / 3}$, and $(b^{1 / 4})^2$ becomes $b^{(1 / 4) * 2} = b^{1 / 2}$. So, our denominator simplifies to $100 a^{-4 / 3} b^{1 / 2}$. Now, our entire expression looks like this: $\frac{5 a^{1 / 3} b^{-2}}{100 a^{-4 / 3} b^{1 / 2}}$. This transformation sets the stage for the next phase, where we'll combine like terms and further simplify the expression. By carefully applying the power rules, we've managed to break down a complex expression into manageable components.

Simplifying the Numerator and Denominator

Now that we've expanded the denominator, the expression stands as $\frac{5 a^{1 / 3} b^{-2}}{100 a^{-4 / 3} b^{1 / 2}}$. The next step involves simplifying the coefficients and dealing with the variables that have exponents. First, let’s simplify the coefficients. We have $\frac{5}{100}$, which reduces to $\frac{1}{20}$. Next, we'll address the variables. When dividing terms with the same base, we subtract the exponents: $\frac{x^m}{x^n} = x^{m-n}$. For the variable 'a', we have $\frac{a^{1 / 3}}{a^{-4 / 3}}$. Subtracting the exponents gives us $a^{(1 / 3) - (-4 / 3)} = a^{(1 / 3) + (4 / 3)} = a^{5 / 3}$. For the variable 'b', we have $\frac{b^{-2}}{b^{1 / 2}}$. Subtracting the exponents gives us $b^{-2 - (1 / 2)} = b^{-5 / 2}$. Now, putting it all together, our expression looks like $\frac{1}{20} a^{5 / 3} b^{-5 / 2}$. We're almost there! We’ve managed to simplify the expression significantly by applying the division rule for exponents and reducing the coefficients. The remaining step is to address the negative exponent in the variable 'b', which we will tackle in the next section. Each step in this process brings us closer to the simplest form of the expression.

Dealing with Negative Exponents

At this stage, our expression is $\frac{1}{20} a^{5 / 3} b^{-5 / 2}$. We notice the negative exponent on the 'b' term, which isn't considered simplified in mathematical convention. To eliminate the negative exponent, we use the rule $x^{-n} = \frac{1}{x^n}$. Therefore, $b^{-5 / 2}$ becomes $\frac{1}{b^{5 / 2}}$. Substituting this back into our expression, we get $\frac{1}{20} a^{5 / 3} * \frac{1}{b^{5 / 2}}$, which can be rewritten as $\frac{a^{5 / 3}}{20 b^{5 / 2}}$. Now, the expression is simplified in terms of exponents, but we can also consider rewriting it using radicals. Remember that a fractional exponent $x^{m/n}$ can be expressed as $\sqrt[n]{x^m}$. Thus, $a^{5 / 3}$ is the same as $\sqrt[3]{a^5}$, and $b^{5 / 2}$ is the same as $\sqrt{b^5}$. So, our expression can also be written as $\frac{\sqrt[3]{a^5}}{20 \sqrt{b^5}}$. While this form is mathematically correct, the form with fractional exponents is often preferred for its conciseness. By addressing the negative exponent and understanding the relationship between fractional exponents and radicals, we've fully simplified the expression. The result, $\frac{a^{5 / 3}}{20 b^{5 / 2}}$, is the most simplified form, showcasing the power of exponent rules in action.

Final Simplified Expression

After meticulously applying the rules of exponents, we've arrived at the final simplified form of the expression $\frac{5 a^{1 / 3} b^{-2}}{\left(10 a^{-2 / 3} b^{1 / 4}\right)^2}$. The simplified expression is $\frac{a^{5 / 3}}{20 b^{5 / 2}}$. This form is not only mathematically correct but also concise, making it easier to work with in further calculations or analyses. Throughout this process, we've utilized key exponent rules such as the power of a product rule, the power of a power rule, and the division rule for exponents. We've also tackled negative exponents and understood their relationship with reciprocals. Moreover, we've seen how fractional exponents connect to radicals, providing us with different ways to represent the same mathematical idea. The journey from the original complex expression to its simplified form highlights the elegance and efficiency of mathematical tools. By mastering these tools, you can confidently simplify a wide array of expressions, making mathematical problem-solving more accessible and enjoyable. Remember, practice is key. The more you work with these rules, the more intuitive they become. Continue to explore, simplify, and appreciate the power of mathematics in action.

Conclusion

Simplifying expressions with fractional and negative exponents might seem daunting at first, but by breaking down the problem step-by-step and applying the fundamental rules of exponents, it becomes a manageable task. We successfully simplified the given expression $\frac{5 a^{1 / 3} b^{-2}}{\left(10 a^{-2 / 3} b^{1 / 4}\right)^2}$ to $\frac{a^{5 / 3}}{20 b^{5 / 2}}$. This process underscores the importance of understanding and applying exponent rules, which are essential tools in algebra and higher mathematics. Remember to tackle each component methodically, addressing parentheses, negative exponents, and fractional exponents in a logical sequence. With practice, these techniques will become second nature, allowing you to confidently approach even the most complex expressions. For further exploration and practice on exponent rules, visit a trusted resource like Khan Academy's Exponents and Radicals to solidify your understanding and enhance your mathematical skills. Keep practicing, and you'll find that simplifying expressions becomes an empowering part of your mathematical journey.