Simplify 3√2 + √3 / 5√6 - √2: A Detailed Guide

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Hey guys! Math can sometimes feel like navigating a maze, right? Especially when you're faced with expressions that look like a jumble of square roots and fractions. But don't worry! Today, we're going to break down one of those seemingly complex problems and make it super easy to understand. We'll tackle the simplification of the expression 3√2 + √3 / 5√6 - √2 step by step. So, grab your pencils, notebooks, and let's dive in!

Understanding the Basics

Before we jump into the main problem, it's essential to have a solid grasp of the fundamental concepts. First off, let's talk about radicals, those funky-looking √ symbols. A radical, or square root, asks the question: “What number, when multiplied by itself, equals the number under the radical sign?” For example, √9 is 3 because 3 * 3 = 9. Radicals can sometimes be simplified if the number under the root has a perfect square as a factor. Like, √12 can be simplified because 12 has a factor of 4, which is a perfect square (2 * 2 = 4). So, √12 becomes √(4 * 3), which simplifies to 2√3. Make sense? Another important concept is rationalizing the denominator. This is a technique we use to get rid of any radicals in the denominator of a fraction. It makes the expression look cleaner and is generally considered good mathematical practice. We achieve this by multiplying both the numerator and denominator by the conjugate of the denominator. The conjugate is simply the same expression with the opposite sign in the middle. For example, the conjugate of a + b is a - b. Why do we do this? Because when you multiply an expression by its conjugate, you eliminate the radical in the denominator, as we'll see in our example. Keep these basics in mind, and you'll be simplifying radicals like a pro in no time! Simplifying radical expressions often involves combining like terms. Like terms are terms that have the same radical part. For instance, 2√3 and 5√3 are like terms because they both have √3. We can add or subtract like terms just like we do with regular variables. So, 2√3 + 5√3 = 7√3. However, 2√3 and 2√2 are not like terms because they have different radicals. You can't directly combine them. When simplifying expressions with radicals, you'll often need to use the distributive property, just like with algebraic expressions. Remember, the distributive property states that a(b + c) = ab + ac. We apply this same principle when multiplying a number or radical by a sum or difference involving radicals. For example, √2(3 + √5) = 3√2 + √10. By understanding and mastering these fundamental concepts, you'll be well-equipped to tackle more complex problems involving radicals and rationalizing denominators. These building blocks are crucial for simplifying expressions and solving equations that involve radicals. With practice, you'll become more comfortable and confident in your ability to manipulate these expressions and arrive at the correct solutions.

Step-by-Step Breakdown of 3√2 + √3 / 5√6 - √2

Okay, let's get to the heart of the problem: simplifying 3√2 + √3 / 5√6 - √2. This might look a bit intimidating at first glance, but trust me, we'll break it down into manageable steps.

  1. Identify the Expression: The expression we're working with is (3√2 + √3) / (5√6 - √2). Notice that we have a fraction with radicals in both the numerator and the denominator. This is where the concept of rationalizing the denominator comes in handy.

  2. Find the Conjugate: Remember, the key to rationalizing the denominator is multiplying by its conjugate. The denominator here is 5√6 - √2. To find its conjugate, we simply change the sign in the middle. So, the conjugate of 5√6 - √2 is 5√6 + √2. This is going to be our magic bullet for getting rid of those pesky radicals in the denominator.

  3. Multiply by the Conjugate: Now, we're going to multiply both the numerator and the denominator of our expression by the conjugate we just found. This is crucial because multiplying the denominator by its conjugate will eliminate the radicals there. So, we have: ((3√2 + √3) / (5√6 - √2)) * ((5√6 + √2) / (5√6 + √2)). Multiplying by the conjugate over itself is the same as multiplying by 1, so we're not changing the value of the expression, just its appearance. This step sets the stage for simplifying the expression and getting rid of the radicals in the denominator.

  4. Expand the Numerator: Let's focus on the numerator first. We need to multiply (3√2 + √3) by (5√6 + √2). This is where the distributive property (or the FOIL method) comes in handy. We'll multiply each term in the first set of parentheses by each term in the second set: (3√2 * 5√6) + (3√2 * √2) + (√3 * 5√6) + (√3 * √2). Now let's simplify each of these terms individually. Remember that √a * √b = √(a * b). So, (3√2 * 5√6) becomes 15√12. (3√2 * √2) becomes 3√4. (√3 * 5√6) becomes 5√18. And (√3 * √2) becomes √6. Our expanded numerator now looks like this: 15√12 + 3√4 + 5√18 + √6. But we're not done yet! We can simplify the radicals further.

  5. Expand the Denominator: Now, let's tackle the denominator. We're multiplying (5√6 - √2) by its conjugate (5√6 + √2). This is where the magic happens! When we multiply conjugates, the middle terms cancel out, leaving us with a simpler expression. Using the distributive property again: (5√6 * 5√6) + (5√6 * √2) - (√2 * 5√6) - (√2 * √2). Notice that the terms (5√6 * √2) and -(√2 * 5√6) are the same but with opposite signs, so they cancel each other out. We're left with: (5√6 * 5√6) - (√2 * √2). Simplifying further: (25 * 6) - 2, which equals 150 - 2, which equals 148. So, our denominator is now a nice, clean 148 – no more radicals!

  6. Simplify Radicals: Remember those radicals in the expanded numerator? Now's the time to simplify them. Let's go back to our expanded numerator: 15√12 + 3√4 + 5√18 + √6. We can simplify √12, √4, and √18. √12 can be written as √(4 * 3), which simplifies to 2√3. √4 is simply 2. √18 can be written as √(9 * 2), which simplifies to 3√2. So, our numerator now looks like this: 15 * 2√3 + 3 * 2 + 5 * 3√2 + √6, which simplifies to 30√3 + 6 + 15√2 + √6. We've made some serious progress here!

  7. Combine Like Terms: Next, we want to combine any like terms in the numerator. Looking at our numerator, 30√3 + 6 + 15√2 + √6, we see that there are no like radical terms that can be combined. The terms √3, √2, and √6 are all different, so we can't add them together directly. The only like terms we could potentially combine are constants, but in this case, we only have one constant term: 6. So, there's nothing to combine in this step. This means our numerator remains as 30√3 + 6 + 15√2 + √6.

  8. Final Simplification: Now, let's put it all together. We have our simplified numerator: 30√3 + 6 + 15√2 + √6, and our simplified denominator: 148. Our expression now looks like this: (30√3 + 6 + 15√2 + √6) / 148. As a final touch, we can see if there's a common factor we can divide out of both the numerator and the denominator to simplify further. Looking at the coefficients in the numerator (30, 6, 15, and 1) and the denominator (148), we can see that the greatest common factor (GCF) is 1. This means we can't simplify the fraction any further by dividing out a common factor. However, we can rewrite the expression to make it look a bit cleaner. We can separate the fraction into individual terms: (30√3 / 148) + (6 / 148) + (15√2 / 148) + (√6 / 148). Now, we can simplify each fraction individually by dividing the coefficients by their greatest common factor, if there is one. For example, 30 and 148 have a common factor of 2, so we can simplify 30√3 / 148 to 15√3 / 74. Similarly, 6 and 148 have a common factor of 2, so we can simplify 6 / 148 to 3 / 74. 15 and 148 don't have any common factors other than 1, so 15√2 / 148 remains as is. And finally, 1 and 148 don't have any common factors, so √6 / 148 remains as is. Our fully simplified expression is now: (15√3 / 74) + (3 / 74) + (15√2 / 148) + (√6 / 148).

Tips and Tricks for Simplifying Radical Expressions

Alright, you've seen the step-by-step process for simplifying a complex radical expression. But let's arm you with some extra tips and tricks to make this process even smoother. Think of these as your secret weapons in the battle against complex math problems.

  • Prime Factorization is Your Friend: When dealing with larger numbers under a radical, prime factorization is your best friend. Break down the number into its prime factors. This will help you identify any perfect square factors that you can take out of the radical. For example, let's say you have √72. Instead of trying to guess a perfect square factor, break 72 down into its prime factors: 2 x 2 x 2 x 3 x 3. You can rewrite this as (2 x 2) x (3 x 3) x 2, which is 2² x 3² x 2. Now you can easily see that √72 = √(2² x 3² x 2) = 2 x 3 x √2 = 6√2. Prime factorization makes it clear and easy!
  • Look for Perfect Squares: Always be on the lookout for perfect square factors (4, 9, 16, 25, etc.) within the radical. Identifying these quickly will speed up your simplification process. For instance, if you spot √48, immediately recognize that 16 is a perfect square factor (48 = 16 x 3). This means √48 = √(16 x 3) = 4√3. The quicker you spot those perfect squares, the faster you'll simplify!
  • Simplify Before You Multiply: When multiplying radicals, it's often easier to simplify each radical first before multiplying. This keeps the numbers smaller and more manageable. For example, if you have √8 * √18, you could multiply them directly to get √144, which simplifies to 12. But it's often easier to simplify first: √8 = 2√2 and √18 = 3√2. Now, multiply the simplified radicals: 2√2 * 3√2 = 6 * 2 = 12. Same answer, but potentially less chance for errors with smaller numbers.
  • Rationalize Carefully: When rationalizing the denominator, double-check that you're multiplying both the numerator and the denominator by the conjugate. It's a common mistake to forget the numerator, which will throw off your entire answer. Also, remember that multiplying by the conjugate is only necessary when the denominator has two terms (like a + √b or √a - √b). If the denominator is a single radical term (like √5), you can simply multiply the numerator and denominator by that radical.
  • Practice Makes Perfect: Like any math skill, simplifying radical expressions gets easier with practice. The more you work through different problems, the more comfortable you'll become with the process and the faster you'll be able to spot patterns and shortcuts. So, don't be afraid to tackle a variety of problems! Work through examples in your textbook, online resources, or practice worksheets. The key is consistent practice.
  • Double-Check Your Work: Always take a moment to double-check your work, especially when dealing with multiple steps. It's easy to make a small mistake, like a sign error or a missed simplification. A quick review can save you from losing points on an exam or assignment. Pay close attention to your arithmetic and make sure you've simplified each radical as much as possible.

Common Mistakes to Avoid

Even with a solid understanding of the steps, it's easy to stumble when simplifying radical expressions. Let's highlight some common pitfalls so you can steer clear of them. Think of this as your