Simplifying The Expression A = (8/√2 - √72 + √128) / (3√6 / 8)
Introduction to Simplifying Radical Expressions
In the realm of mathematics, simplifying radical expressions is a fundamental skill that often appears in various algebraic problems. These expressions, which involve square roots, cube roots, and higher-order roots, can initially seem daunting. However, by understanding the properties of radicals and applying systematic simplification techniques, one can efficiently reduce these expressions to their simplest forms. This not only makes the expressions easier to work with but also provides a deeper understanding of their underlying values. This article delves into the process of simplifying radical expressions, focusing on the specific example: a = (8/√2 - √72 + √128) : (3√6 / 8). We will break down each step involved in simplifying this expression, highlighting the key mathematical principles at play.
Breaking Down the Expression: a = (8/√2 - √72 + √128) : (3√6 / 8)
The expression a = (8/√2 - √72 + √128) : (3√6 / 8) appears complex at first glance, but we can simplify it by addressing each term individually. The expression involves a combination of rationalizing denominators, simplifying square roots, and performing arithmetic operations. Let's dissect this expression step by step to make the simplification process clear and understandable. Firstly, we will focus on rationalizing the denominator of the first term and then proceed to simplify the square roots in the subsequent terms. Understanding each component will allow us to methodically reduce the expression to its simplest form. This methodical approach is crucial for tackling more complex mathematical problems.
Rationalizing the Denominator: 8/√2
The first term in our expression is 8/√2. In mathematics, it's common practice to rationalize the denominator, which means eliminating the radical from the denominator. To achieve this, we multiply both the numerator and the denominator by the radical present in the denominator, which in this case is √2. This process ensures that we are not changing the value of the expression, as we are essentially multiplying by 1. The operation transforms the term into (8 * √2) / (√2 * √2). Simplifying further, we get 8√2 / 2. Finally, we can reduce the fraction by dividing both the numerator and the denominator by 2, resulting in 4√2. This simplification is a critical step in handling expressions with radicals in the denominator.
Simplifying √72
Next, we need to simplify √72. To do this, we look for the largest perfect square that divides 72. A perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25, etc.). In this case, the largest perfect square that divides 72 is 36, since 72 = 36 * 2. Thus, we can rewrite √72 as √(36 * 2). Using the property of radicals that √(ab) = √a * √b, we can separate this into √36 * √2. Since √36 = 6, the expression simplifies to 6√2. This method of finding perfect square factors is essential for simplifying square roots.
Simplifying √128
Now, let's simplify √128. Similar to the previous step, we need to find the largest perfect square that divides 128. The largest perfect square in this case is 64, as 128 = 64 * 2. Therefore, we can rewrite √128 as √(64 * 2). Applying the property √(ab) = √a * √b, we separate this into √64 * √2. Since √64 = 8, the expression simplifies to 8√2. Recognizing and extracting perfect square factors is a key technique in simplifying radical expressions.
Combining Simplified Terms
After simplifying each radical term individually, we can now combine them. Our original expression (8/√2 - √72 + √128) has been transformed into (4√2 - 6√2 + 8√2). These terms are now like terms because they all contain the same radical, √2. We can combine like terms by simply adding or subtracting their coefficients. In this case, we have 4√2 - 6√2 + 8√2. Adding the coefficients, we get (4 - 6 + 8)√2, which simplifies to 6√2. This step demonstrates the importance of simplifying individual terms before attempting to combine them.
Dividing by (3√6 / 8)
The next part of our original expression is the division by (3√6 / 8). Recall that dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we need to find the reciprocal of (3√6 / 8), which is 8 / (3√6). Now, we multiply our simplified expression 6√2 by 8 / (3√6). This gives us (6√2) * (8 / (3√6)), which equals (48√2) / (3√6). To further simplify this, we can divide 48 by 3, resulting in (16√2) / √6. Simplifying this fraction involves dealing with radicals in both the numerator and the denominator.
Further Simplification: (16√2) / √6
To further simplify (16√2) / √6, we need to rationalize the denominator again. We multiply both the numerator and the denominator by √6. This gives us (16√2 * √6) / (√6 * √6). Simplifying the denominator, we get (16√2 * √6) / 6. Now, we can simplify the numerator by multiplying the radicals: √2 * √6 = √12. So, we have (16√12) / 6. Next, we simplify √12. The largest perfect square that divides 12 is 4, so we can rewrite √12 as √(4 * 3), which simplifies to 2√3. Substituting this back into our expression, we get (16 * 2√3) / 6, which equals (32√3) / 6. Finally, we can reduce the fraction by dividing both the numerator and the denominator by 2, resulting in (16√3) / 3. This is the simplest form of our expression.
Final Solution and Summary
In conclusion, by systematically simplifying each term and applying the rules of radical expressions, we have found that the simplified form of a = (8/√2 - √72 + √128) : (3√6 / 8) is (16√3) / 3. This process involved rationalizing denominators, simplifying square roots by identifying perfect square factors, combining like terms, and multiplying by reciprocals. The ability to simplify radical expressions is a crucial skill in algebra and is essential for solving more complex mathematical problems. The step-by-step approach demonstrated here can be applied to a wide range of similar problems, making it a valuable tool for any student of mathematics.
Common Mistakes to Avoid
When simplifying radical expressions, there are several common mistakes students often make. Understanding these pitfalls can help you avoid them and improve your accuracy. One frequent error is failing to completely simplify a radical. For instance, if you encounter √20, you might stop at √(4 * 5) without further simplifying to 2√5. Always ensure that you have extracted the largest possible perfect square factor. Another mistake is incorrectly combining terms that are not like terms. For example, 3√2 + 2√3 cannot be combined because the radicals are different. You can only combine terms with the same radical. Additionally, errors can occur during rationalization of denominators. Remember to multiply both the numerator and the denominator by the correct radical to eliminate the radical from the denominator. Paying close attention to these common pitfalls can significantly enhance your ability to simplify radical expressions correctly.
Practice Problems for Mastering Radical Simplification
To truly master the simplification of radical expressions, practice is essential. Working through a variety of problems will solidify your understanding of the techniques and help you develop problem-solving skills. Here are a few practice problems you can try:
- Simplify: √(48) - √(27) + 2√(12)
- Simplify: (5/√3) + √(75) - √(12)
- Simplify: (√80 / √5) + √(45) - 2√5
- Simplify: (3√2 + √18 - √32) / √2
Working through these problems will not only improve your proficiency but also deepen your understanding of the underlying mathematical principles. Remember to break each problem down into smaller steps, and always double-check your work to ensure accuracy. The more you practice, the more confident and skilled you will become in simplifying radical expressions.