Simplifying $3√3 × 2√2$: A Step-by-Step Guide

Are you struggling with simplifying expressions involving square roots? You're not alone! Many students find these types of problems tricky, but with a clear understanding of the rules and a step-by-step approach, you can master them. This guide will walk you through how to simplify the expression $3 \sqrt{3} \times 2 \sqrt{2}$, breaking down each step to ensure you grasp the underlying concepts. Let's dive in and conquer this mathematical challenge together!

Understanding the Basics of Simplifying Radicals

Before we tackle the main problem, it's crucial to understand the fundamental principles of simplifying radicals. Radicals, often represented by the square root symbol ($\sqrt{}$), indicate the root of a number. Simplifying radicals involves expressing them in their simplest form, where the number under the radical (the radicand) has no perfect square factors other than 1. This process often involves using the product property of radicals, which states that $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$, where a and b are non-negative numbers. Mastering this property is key to simplifying more complex expressions. Also, keep in mind that multiplying radicals involves multiplying the coefficients (the numbers outside the radical) and the radicands separately. This foundational knowledge will be invaluable as we proceed with simplifying the given expression. Remember, practice makes perfect, so don't hesitate to review these basics and work through additional examples to solidify your understanding. By building a strong foundation, you'll be well-equipped to handle any radical simplification problem that comes your way. This understanding is the cornerstone of simplifying radicals, and with consistent effort, you'll find these types of problems becoming much more manageable. Furthermore, recognizing perfect squares within the radicand is crucial for simplification. For instance, knowing that 9 is a perfect square (3 x 3) allows you to simplify \sqrt{9} to 3. This skill will significantly streamline the simplification process, especially in more complex expressions.

Step-by-Step Solution for $3√3 × 2√2$

Now, let's apply these principles to simplify the expression $3 \sqrt{3} \times 2 \sqrt{2}$. We'll break down the process into manageable steps to make it easy to follow. Firstly, we'll use the commutative and associative properties of multiplication to rearrange and regroup the terms. This means we can multiply the coefficients (the numbers outside the square roots) together and the radicands (the numbers inside the square roots) together separately. This initial step sets the stage for a clearer and more organized simplification. Next, we multiply the coefficients: 3 multiplied by 2 equals 6. Then, we multiply the radicands: 3 multiplied by 2 equals 6. So, now we have $6\sqrt{6}$. This is a crucial step in simplifying expressions involving radicals. Finally, we examine the radicand (6) to see if it has any perfect square factors other than 1. In this case, 6 can be factored as 2 x 3, and neither 2 nor 3 are perfect squares. Therefore, $\sqrt{6}$ is already in its simplest form. This final check ensures that our answer is fully simplified and cannot be reduced further. By following these steps systematically, you can confidently simplify similar expressions and avoid common pitfalls. Remember to always double-check your work and ensure that the radicand contains no perfect square factors. With practice, this process will become second nature, and you'll be able to simplify radicals with ease.

Detailed Breakdown of Each Step

To ensure clarity, let's dissect each step in detail. The expression we're tackling is $3 \sqrt{3} \times 2 \sqrt{2}$.

• Step 1: Rearrange and Regroup:

  We begin by using the commutative and associative properties of multiplication to rearrange the terms. This allows us to group the coefficients together and the radicands together. So, we rewrite the expression as $(3 \times 2) \times (\sqrt{3} \times \sqrt{2})$. This step is vital for organizing the expression and making the subsequent steps more straightforward. By clearly separating the coefficients and the radicals, we minimize the chance of errors and ensure a smoother simplification process. Think of it as sorting your tools before starting a project; it makes everything more efficient and less confusing. The commutative property allows us to change the order of multiplication without affecting the result (a x b = b x a), and the associative property allows us to change the grouping of factors without affecting the result ( (a x b) x c = a x (b x c) ). Applying these properties is a fundamental technique in simplifying mathematical expressions.

• Step 2: Multiply the Coefficients:

  Next, we multiply the coefficients, which are the numbers outside the square roots. In this case, we multiply 3 by 2, which gives us 6. So, our expression now looks like $6 \times (\sqrt{3} \times \sqrt{2})$. Multiplying the coefficients is a straightforward arithmetic operation, but it's an essential part of the simplification process. This step combines the numerical components of the expression, paving the way for simplifying the radical part. Remember to pay close attention to the signs of the coefficients, as this can affect the final result. If any of the coefficients were negative, we would need to account for that in this step. For example, if we had -3 instead of 3, the result would be -6. Accuracy in this step is crucial for arriving at the correct simplified expression.

• Step 3: Multiply the Radicands:

  Now, we multiply the radicands, which are the numbers inside the square roots. We multiply 3 by 2, which equals 6. This gives us $\sqrt{6}$. Now our expression is $6\sqrt{6}$. This step utilizes the product property of radicals, which states that the square root of a product is equal to the product of the square roots. By multiplying the radicands, we consolidate the radical part of the expression into a single square root. This simplifies the expression further and makes it easier to determine if any further simplification is possible. It's important to note that this step only applies to radicals with the same index (in this case, square roots). We couldn't directly multiply the radicands if one was a square root and the other was a cube root, for instance. Understanding this limitation is key to applying the product property of radicals correctly.

• Step 4: Simplify the Resulting Radical:

  Finally, we check if the radicand, 6, can be simplified further. We look for perfect square factors of 6. The factors of 6 are 1, 2, 3, and 6. None of these, other than 1, are perfect squares (perfect squares are numbers like 4, 9, 16, etc., which are the result of squaring an integer). Therefore, $\sqrt{6}$ is already in its simplest form. This final step is crucial to ensure that the expression is fully simplified. Failing to check for perfect square factors can lead to an incomplete simplification. If we had a radicand like 12, for example, we would need to recognize that 12 has a perfect square factor of 4 (12 = 4 x 3) and simplify $\sqrt{12}$ to $2\sqrt{3}$. Since 6 has no perfect square factors, we can confidently conclude that $6\sqrt{6}$ is the simplest form of the expression.

The Final Answer

Therefore, the simplified form of $3 \sqrt{3} \times 2 \sqrt{2}$ is $6 \sqrt{6}$. This matches option A in your provided choices. By following these steps, you can confidently simplify similar expressions involving radicals. Remember, practice is key! The more you work with these types of problems, the more comfortable and proficient you'll become. Don't be afraid to tackle challenging expressions and break them down into smaller, manageable steps. With a solid understanding of the underlying principles and a systematic approach, you can conquer any radical simplification problem.

Common Mistakes to Avoid

When simplifying radicals, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accurate solutions. One frequent error is forgetting to multiply both the coefficients and the radicands. Remember, you need to multiply the numbers outside the square roots and the numbers inside the square roots separately. Another mistake is failing to simplify the radical completely. Always check if the radicand has any perfect square factors that can be extracted. For instance, if you end up with $\sqrt{8}$, you should simplify it to $2\sqrt{2}$ because 8 has a perfect square factor of 4. Additionally, some students mistakenly try to add or subtract radicands that are not like terms. You can only add or subtract radicals if they have the same radicand. For example, $2\sqrt{3} + 3\sqrt{3}$ can be simplified to $5\sqrt{3}$, but $2\sqrt{3} + 3\sqrt{2}$ cannot be simplified further. Another common mistake is incorrectly applying the product property of radicals. Remember that $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$, but this property does not apply to addition or subtraction. $\sqrt{a + b}$ is not equal to $\sqrt{a} + \sqrt{b}$. By being mindful of these common errors, you can improve your accuracy and simplify radicals with confidence.

Practice Problems

To solidify your understanding, try simplifying these expressions:

1. $4 \sqrt{5} \times 3 \sqrt{2}$
2. $2 \sqrt{7} \times 5 \sqrt{7}$
3. $\sqrt{18} \times \sqrt{2}$
4. $3 \sqrt{8} \times 2 \sqrt{3}$

Working through these practice problems will help you reinforce the steps we've discussed and identify any areas where you may need further clarification. Remember to break down each problem into smaller steps, focusing on multiplying the coefficients, multiplying the radicands, and then simplifying the resulting radical. Don't hesitate to refer back to the step-by-step guide if you encounter any difficulties. The key to mastering radical simplification is consistent practice and a thorough understanding of the underlying principles. As you work through these problems, pay attention to the common mistakes we discussed earlier and actively try to avoid them. With each problem you solve, you'll gain more confidence and proficiency in simplifying radicals.

Conclusion

Simplifying radical expressions like $3 \sqrt{3} \times 2 \sqrt{2}$ might seem daunting at first, but by breaking it down into manageable steps, it becomes a straightforward process. Remember to multiply the coefficients, multiply the radicands, and then simplify the resulting radical. Always look for perfect square factors within the radicand to ensure you've simplified the expression completely. Consistent practice and a clear understanding of the rules will empower you to tackle any radical simplification problem with confidence. Keep practicing, and you'll become a pro at simplifying radicals in no time! For further learning and practice, check out this helpful resource on Khan Academy. Good luck, and happy simplifying!