Calculating Radicals: Step-by-Step Solutions

Hey guys! Today, we're diving deep into the world of radicals and tackling some tricky calculations. If you've ever felt lost trying to simplify expressions with square roots, you're in the right place. We're going to break down two problems step-by-step, so you'll not only get the answers but also understand the how and why behind each step. So, grab your pencils, and let's get started!

Part a) √(18 + 2√8 - 5√50)

In this section, we will calculate the value of the expression √(18 + 2√8 - 5√50). Mastering radical simplification is crucial in algebra and calculus, and this example offers a comprehensive approach to tackling such problems. Our primary goal is to simplify the expression inside the square root by expressing each term in its simplest radical form. This involves identifying perfect square factors within the radicands (the numbers inside the square root) and extracting them. By doing so, we aim to combine like terms, which will eventually lead to a single, simplified radical expression. This process requires a good understanding of the properties of square roots and how they interact with arithmetic operations. Let’s dive in and break down the steps involved in simplifying this expression.

Step 1: Simplify Individual Radicals

Alright, let's kick things off by simplifying each radical term individually. This is a fundamental step when you're dealing with radical expressions because it helps to break down complex terms into more manageable ones. We're essentially trying to find perfect square factors within each radical. Think of it like this: we want to rewrite the numbers inside the square roots as products of perfect squares and other numbers. This is crucial because the square root of a perfect square is a whole number, which helps us simplify the overall expression. By focusing on simplifying each radical term separately, we make the entire calculation process much smoother and less prone to errors. Let's look at how this works in practice with our specific problem.

• √8: We can rewrite 8 as 4 * 2. Since 4 is a perfect square (2 * 2 = 4), we can simplify √8 as √(4 * 2) = √4 * √2 = 2√2.
• √50: Similarly, we can rewrite 50 as 25 * 2. Here, 25 is a perfect square (5 * 5 = 25), so √50 becomes √(25 * 2) = √25 * √2 = 5√2.

Step 2: Substitute Simplified Radicals

Now that we've broken down √8 and √50 into their simplest forms, it's time to plug these simplified expressions back into the original equation. This is a super important step because it allows us to rewrite the entire expression in terms of simpler radicals, making it easier to combine like terms. By substituting these values, we transform the original problem into a form that's much more manageable. Think of it as replacing complex pieces of a puzzle with simpler ones that fit together more easily. This substitution is the key to unlocking the next stage of simplification, where we'll be able to combine similar radical terms and further reduce the expression. Let's see how this substitution plays out in our calculation.

Substituting these simplified radicals into the original expression, we get:

√(18 + 2√8 - 5√50) = √(18 + 2(2√2) - 5(5√2))

Step 3: Simplify Inside the Square Root

Okay, we've substituted the simplified radicals, and now it's time to clean things up inside the square root. This step is all about performing the multiplication and combining like terms to make the expression as neat as possible. Think of it as tidying up before you can move on to the final solution. We need to deal with any coefficients multiplying the radicals and then see if we can combine any terms that have the same radical part. This is essential because simplifying the inside of the square root will help us see the expression more clearly and potentially identify further simplifications. By carefully performing these arithmetic operations, we're paving the way for extracting the square root and arriving at our final answer. Let's get to it and see what we can simplify.

Let's simplify the expression inside the square root:

√(18 + 2(2√2) - 5(5√2)) = √(18 + 4√2 - 25√2)

Now, combine the terms with √2:

√(18 + 4√2 - 25√2) = √(18 - 21√2)

Step 4: Final Calculation

Well, guys, it looks like we've hit a bit of a snag! √(18 - 21√2) doesn't simplify to a neat whole number or a simple radical. Sometimes, expressions just don't have a clean simplification, and that's totally okay. It's important to recognize when we've gone as far as we can with simplification. In this case, we've simplified the radicals individually, substituted them back into the expression, and combined like terms. We've done all the standard simplification techniques. So, while we might not have a single, elegant answer, we've still made progress by getting the expression into its simplest form. Sometimes, the most simplified form is still a bit complex, and that's just the nature of some mathematical problems.

Thus, the final simplified expression is √(18 - 21√2).

Part b) 40√32 : (8√8)

In this section, we're tackling the division of radical expressions, specifically 40√32 : (8√8). This type of problem is common in algebra and requires a solid understanding of how to simplify and manipulate radicals, especially when they're involved in division. Our goal here is to simplify both the numerator and the denominator as much as possible before performing the division. This often involves breaking down the numbers inside the square roots into their prime factors and looking for perfect square factors that can be extracted. Once we've simplified the radicals, we can then simplify the numerical coefficients and, if necessary, rationalize the denominator. This comprehensive approach ensures that we arrive at the simplest form of the expression. Let's break down each step and see how we can simplify this division problem.

Step 1: Simplify Individual Radicals

Alright, just like in the first problem, the first step here is to simplify the radicals individually. This is key to making the division process smoother and more manageable. Simplifying radicals involves breaking down the numbers inside the square roots (the radicands) into their prime factors and identifying any perfect square factors. Remember, a perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25, etc.). When we find these perfect square factors, we can take their square roots and move them outside the radical sign. This not only simplifies the radicals but also makes it easier to see how the terms relate to each other, which is especially helpful when dividing radical expressions. Let's dive in and see how we can simplify √32 and √8.

• √32: We can rewrite 32 as 16 * 2. Since 16 is a perfect square (4 * 4 = 16), √32 becomes √(16 * 2) = √16 * √2 = 4√2.
• √8: As we found earlier, 8 can be rewritten as 4 * 2, where 4 is a perfect square. Thus, √8 simplifies to √(4 * 2) = √4 * √2 = 2√2.

Step 2: Substitute Simplified Radicals

Now that we've simplified √32 and √8, it's time to plug these simplified forms back into our original division problem. This is a crucial step in making the expression easier to work with. By replacing the original radicals with their simplified forms, we reduce the complexity of the expression and make it clearer how the different parts relate to each other. Think of it like swapping out clunky, hard-to-handle pieces for sleek, streamlined ones. This substitution allows us to see the underlying structure of the expression more clearly, setting the stage for the next steps of simplification, where we'll focus on dividing the coefficients and simplifying any remaining radical terms. Let's see how this substitution transforms our division problem.

Substituting these simplified radicals into the original expression, we get:

40√32 : (8√8) = 40(4√2) : (8(2√2))

Step 3: Simplify the Expression

Alright, we've substituted the simplified radicals, and now it's time to simplify the whole expression. This step is all about cleaning things up and making the expression as neat as possible. We'll start by performing the multiplications in both the numerator and the denominator. This will give us a clearer picture of the numerical coefficients and the radical terms. Then, we'll look for opportunities to cancel out common factors and simplify the fraction. This might involve dividing both the numerator and the denominator by a common number or simplifying the radical terms themselves. The goal here is to reduce the expression to its simplest form, which means getting rid of any unnecessary clutter and making the numbers as small as possible. Let's dive in and see how we can simplify this expression.

Let's perform the multiplication:

40(4√2) : (8(2√2)) = 160√2 : (16√2)

Now, we can rewrite the division as a fraction:

160√2 / (16√2)

Notice that √2 appears in both the numerator and the denominator, so we can cancel it out:

160√2 / (16√2) = 160 / 16

Step 4: Final Calculation

We're in the home stretch now! We've simplified the radicals, substituted them back into the expression, and canceled out common factors. All that's left is to perform the final division. This is the satisfying moment where we get to see the result of all our hard work. We have a simple fraction, 160 / 16, and we just need to divide the numerator by the denominator to get our final answer. This step is straightforward, but it's crucial to get it right to ensure we have the correct solution. So, let's do this final calculation and wrap up this problem.

Finally, divide 160 by 16:

160 / 16 = 10

Conclusion

So, guys, we've successfully calculated both parts of our problem! For part a), we simplified √(18 + 2√8 - 5√50) to √(18 - 21√2). And for part b), we found that 40√32 : (8√8) equals 10. These problems show how important it is to break down complex expressions into smaller, manageable steps. Remember, simplifying radicals, substituting, and combining like terms are your best friends when tackling these kinds of questions. Keep practicing, and you'll become a radical-solving pro in no time! If you have any questions or want to try more examples, just let me know. Keep up the great work!