Simplify And Rationalize The Radical Expression -√(75a⁵/b³)

Radical expressions can often seem daunting, especially when they involve fractions and variables. However, by understanding the fundamental principles of simplification and rationalization, you can confidently tackle these expressions. This guide will provide a comprehensive, step-by-step approach to simplifying radical expressions and rationalizing denominators, assuming all variables represent positive real numbers. Let's delve into the intricacies of radical expressions, ensuring a clear understanding of each step involved.

Understanding Radical Expressions

At its core, understanding radical expressions is crucial for simplifying and manipulating them effectively. A radical expression consists of a radical symbol (√), a radicand (the expression under the radical), and an index (the small number indicating the root to be taken). For example, in the expression √(9), the radical symbol is '√', the radicand is '9', and the index is '2' (since it's a square root). In the expression ∛(8), the radical symbol is '∛', the radicand is '8', and the index is '3' (indicating a cube root).

The key to simplifying radical expressions lies in identifying perfect squares, cubes, or higher powers within the radicand. A perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16). Similarly, a perfect cube is a number that can be obtained by cubing an integer (e.g., 8, 27, 64). Recognizing these perfect powers allows us to extract them from the radical, thus simplifying the expression. For instance, √25 can be simplified to 5 because 25 is a perfect square (5² = 25). Similarly, ∛27 simplifies to 3 because 27 is a perfect cube (3³ = 27).

When dealing with variables under a radical, the same principle applies. If the exponent of the variable is divisible by the index of the radical, we can simplify it. For example, √(x²) simplifies to x because the exponent 2 is divisible by the index 2. Similarly, ∛(y³) simplifies to y because the exponent 3 is divisible by the index 3. When the exponent is not perfectly divisible by the index, we look for the largest multiple of the index that is less than or equal to the exponent. For instance, to simplify √(x⁵), we recognize that x⁵ can be written as x⁴ * x, where x⁴ is a perfect square (x²)². Thus, √(x⁵) simplifies to x²√(x).

In addition to understanding perfect powers, the properties of radicals are essential. The product property of radicals states that the square root of a product is equal to the product of the square roots: √(ab) = √a * √b. Similarly, the quotient property states that the square root of a quotient is equal to the quotient of the square roots: √(a/b) = √a / √b. These properties are instrumental in breaking down complex radicals into simpler components. For example, to simplify √48, we can rewrite 48 as 16 * 3, where 16 is a perfect square. Then, using the product property, √48 = √(16 * 3) = √16 * √3 = 4√3. Similarly, to simplify √(25/9), we can use the quotient property: √(25/9) = √25 / √9 = 5/3.

Before diving into more complex scenarios, mastering these fundamental concepts and properties is crucial. Recognizing perfect powers and applying the product and quotient properties form the foundation for simplifying a wide range of radical expressions. In the subsequent sections, we will build upon this foundation by addressing expressions with fractions and rationalizing denominators.

Simplifying Radicals with Fractions

When simplifying radicals with fractions, the initial step involves applying the quotient property of radicals. This property, as mentioned earlier, allows us to separate the radical of a fraction into the fraction of radicals: √(a/b) = √a / √b. This separation is often the key to simplifying complex expressions. For instance, consider the expression √ (75a⁵ / b³). Applying the quotient property, we can rewrite it as √ (75a⁵) / √ (b³). This initial separation allows us to focus on simplifying the numerator and the denominator independently, making the entire process more manageable.

After separating the radical, the next step is to simplify each radical individually. In the numerator, √ (75a⁵), we need to identify perfect square factors. We can break down 75 into 25 * 3, where 25 is a perfect square. Similarly, a⁵ can be written as a⁴ * a, where a⁴ is a perfect square (a²)². Applying these decompositions, √ (75a⁵) becomes √ (25 * 3 * a⁴ * a). Using the product property of radicals, we further separate this into √25 * √3 * √ (a⁴) * √a. Now, we can simplify the perfect squares: √25 = 5 and √ (a⁴) = a². This leaves us with 5 * √3 * a² * √a, which can be written more concisely as 5a²√ (3a). Thus, the simplified form of the numerator is 5a²√ (3a).

Turning our attention to the denominator, √ (b³), we follow a similar approach. We need to identify the largest perfect square factor within b³. Since b³ can be written as b² * b, where b² is a perfect square, we rewrite √ (b³) as √ (b² * b). Applying the product property, this becomes √ (b²) * √b. Simplifying the perfect square, √ (b²) = b, and we are left with b√b. Therefore, the simplified form of the denominator is b√b.

Now that we have simplified both the numerator and the denominator, we can rewrite the original expression with these simplified components. The expression √ (75a⁵ / b³) has now been transformed into (5a²√ (3a)) / (b√b). At this stage, the radical is simplified as much as possible within the numerator and the denominator. However, we are not yet done, particularly if the expression requires rationalizing the denominator, which will be addressed in the next section.

Before moving on, it's essential to reiterate the critical steps in simplifying radicals with fractions. First, apply the quotient property to separate the radical. Second, simplify the numerator and the denominator independently by identifying and extracting perfect square factors. This methodical approach ensures that radicals are simplified effectively, setting the stage for further manipulations such as rationalizing the denominator. By mastering these techniques, one can confidently navigate complex radical expressions involving fractions.

Rationalizing the Denominator

Rationalizing the denominator is a crucial step in simplifying radical expressions, particularly when the denominator contains a radical. The goal of rationalization is to eliminate the radical from the denominator without changing the value of the expression. This is typically achieved by multiplying both the numerator and the denominator by a suitable factor that will transform the denominator into a rational number. When we left off in the previous section, we had the simplified expression (5a²√ (3a)) / (b√b). Now, we need to rationalize the denominator, which currently contains √b.

The key to rationalizing a denominator containing a square root is to multiply both the numerator and the denominator by that square root. In this case, we multiply by √b. The rationale behind this approach is that √b * √b equals b, which is a rational number. By performing this multiplication, we effectively eliminate the square root from the denominator. So, we multiply both the numerator and the denominator of (5a²√ (3a)) / (b√b) by √b. This gives us (5a²√ (3a) * √b) / (b√b * √b).

Now, let’s simplify the new expression. In the numerator, we have 5a²√ (3a) * √b. Using the product property of radicals, we can combine the square roots: √ (3a) * √b = √ (3ab). Thus, the numerator becomes 5a²√ (3ab). In the denominator, we have b√b * √b. As we established earlier, √b * √b = b. Therefore, the denominator simplifies to b * b, which is b². Our expression now looks like (5a²√ (3ab)) / b².

At this point, we have successfully rationalized the denominator. The radical is no longer present in the denominator; it has been transformed into a rational number, b². However, it's essential to examine the expression further to see if any additional simplifications are possible. In this case, we check for common factors between the numerator and the denominator. Here, 5a² and b² do not share any common factors, and the radical in the numerator, √ (3ab), cannot be simplified further since 3, a, and b are distinct factors under the square root.

Thus, the fully simplified and rationalized form of the expression is (5a²√ (3ab)) / b². This final step demonstrates the complete process of simplifying a radical expression with a fraction and rationalizing the denominator. The key takeaways here are to multiply the numerator and the denominator by the radical in the denominator, simplify the resulting expression, and check for any additional simplifications.

Rationalizing the denominator is not just a mathematical exercise; it's a practice that often leads to simpler and more manageable expressions, particularly in further calculations or when comparing different expressions. By mastering this technique, one gains a significant advantage in handling algebraic manipulations involving radicals.

Applying the Steps to the Given Problem

Now, let's apply the simplification and rationalization steps discussed to the given problem: -√ (75a⁵ / b³). This exercise will reinforce the concepts and techniques we've covered, providing a clear example of how to tackle such expressions. The problem requires us to first simplify the radical expression and then rationalize the denominator, assuming that all variables represent positive real numbers. The initial expression presents a combination of numerical coefficients, variables, and a fraction under the radical, making it an ideal candidate for our step-by-step approach.

Starting with the given expression, -√ (75a⁵ / b³), our first step is to apply the quotient property of radicals. This separates the radical of the fraction into the fraction of radicals: - [√ (75a⁵) / √ (b³)]. The negative sign outside the radical remains unaffected by this separation, acting as a multiplier for the entire simplified expression. This separation allows us to focus on simplifying the numerator and the denominator independently, streamlining the process and minimizing potential errors.

Next, we simplify the numerator, √ (75a⁵). As we discussed earlier, identifying perfect square factors is crucial. We break down 75 into 25 * 3, where 25 is a perfect square, and a⁵ into a⁴ * a, where a⁴ is also a perfect square. Thus, √ (75a⁵) can be rewritten as √ (25 * 3 * a⁴ * a). Applying the product property of radicals, we separate this into √25 * √3 * √ (a⁴) * √a. Simplifying the perfect squares, √25 = 5 and √ (a⁴) = a², we get 5 * √3 * a² * √a, which simplifies to 5a²√ (3a). Therefore, the simplified form of the numerator is 5a²√ (3a).

Now, we turn our attention to the denominator, √ (b³). We identify the perfect square factor within b³, which is b². So, we rewrite √ (b³) as √ (b² * b). Applying the product property, this becomes √ (b²) * √b. Simplifying the perfect square, √ (b²) = b, and we are left with b√b. Therefore, the simplified form of the denominator is b√b. Putting the simplified numerator and denominator together, the expression becomes - [5a²√ (3a) / (b√b)].

Having simplified the radical, our next task is to rationalize the denominator. The denominator currently contains √b, so we multiply both the numerator and the denominator by √b. This gives us - [(5a²√ (3a) * √b) / (b√b * √b)]. Simplifying the numerator, we combine the radicals: √ (3a) * √b = √ (3ab), so the numerator becomes 5a²√ (3ab). In the denominator, √b * √b = b, so the denominator simplifies to b * b, which is b². The expression now looks like - [5a²√ (3ab) / b²].

Finally, we check for any additional simplifications. In this case, there are no common factors between the numerator and the denominator, and the radical in the numerator, √ (3ab), cannot be simplified further. Thus, the fully simplified and rationalized form of the given expression -√ (75a⁵ / b³) is - (5a²√ (3ab)) / b². This final answer encapsulates the step-by-step application of simplifying radicals with fractions and rationalizing the denominator, demonstrating a systematic approach to solving such problems.

Conclusion

In conclusion, simplifying radical expressions and rationalizing denominators are fundamental skills in algebra. By breaking down complex problems into manageable steps, we can confidently navigate these challenges. The process involves understanding the properties of radicals, identifying perfect square factors, and strategically multiplying by appropriate factors to eliminate radicals from the denominator. Through this comprehensive guide, we have demonstrated a clear, step-by-step approach, ensuring that even the most intricate radical expressions can be simplified effectively. This mastery not only enhances one's algebraic proficiency but also provides a solid foundation for more advanced mathematical concepts.