Finding The Equivalent Number Of 5√8 ⋅ 3√4 A Step-by-Step Guide

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Hey guys! Let's dive into a fun math problem today. We're going to figure out which number is equivalent to $5 \sqrt{8} \cdot 3 \sqrt{4}$. This might seem a bit tricky at first, but don't worry, we'll break it down step by step. Our goal is to make sure you not only get the answer but also understand why it's the answer. So, let's get started!

Understanding the Problem

Before we jump into solving, let's make sure we understand what the problem is asking. We have the expression $5 \sqrt{8} \cdot 3 \sqrt{4}$, and we need to simplify it to match one of the given options. The options are:

  • A. $15 \sqrt{2}$
  • B. $60 \sqrt{2}$
  • C. $30 \sqrt{3}$
  • D. $60 \sqrt{3}$

So, our main keyword here is equivalent number, and we're dealing with square roots. Keep these in mind as we go through the solution. We're dealing with mathematical expressions that involve both integers and radicals, so a good grasp of these concepts is essential.

Initial Assessment

Okay, so let's look at our initial expression: $5 \sqrt{8} \cdot 3 \sqrt{4}$. The first thing we should do is identify the different parts of the expression. We have integers (5 and 3) and square roots ($\sqrt{8}$ and $\sqrt{4}$). Our plan is to simplify the square roots first, then multiply everything together. This approach breaks down the problem into smaller, more manageable steps, which is super helpful when dealing with more complex math.

Remember, the key here is to simplify. We want to make the numbers under the square root as small as possible. This often involves finding perfect squares that are factors of the numbers inside the square root. For example, 8 can be written as 4 * 2, and 4 is a perfect square. Spotting these perfect squares is crucial for simplifying radicals effectively.

Breaking Down the Square Roots

Let's start with $\sqrt{8}$. We need to find a perfect square that divides 8. As mentioned earlier, 8 can be written as 4 * 2, and 4 is a perfect square (2 * 2 = 4). So, we can rewrite $\sqrt{8}$ as $\sqrt{4 imes 2}$. Using the property of square roots that says $\sqrt{a imes b} = \sqrt{a} imes \sqrt{b}$, we get:

8=4imes2=4imes2=22\sqrt{8} = \sqrt{4 imes 2} = \sqrt{4} imes \sqrt{2} = 2 \sqrt{2}

So, we've simplified $\sqrt{8}$ to $2 \sqrt{2}$. Great job! Now, let's move on to the next square root, $\sqrt{4}$. This one is much simpler because 4 is already a perfect square. The square root of 4 is simply 2. So, $\sqrt{4} = 2$.

Substituting the Simplified Roots

Now that we've simplified both square roots, let's substitute them back into our original expression. We had $5 \sqrt{8} \cdot 3 \sqrt{4}$. Replacing $\sqrt{8}$ with $2 \sqrt{2}$ and $\sqrt{4}$ with 2, we get:

5imes(22)imes3imes25 imes (2 \sqrt{2}) imes 3 imes 2

This looks much more manageable, doesn't it? We've gotten rid of the more complex square roots and now we just have a series of multiplications to perform. Remember, the order of operations (PEMDAS/BODMAS) tells us to do multiplication from left to right.

Performing the Calculation

Now comes the fun part – the actual calculation! We have: $5 imes (2 \sqrt{2}) imes 3 imes 2$. Let's multiply the integers together first. We have 5, 2, 3, and 2. Multiplying these gives us:

5imes2imes3imes2=605 imes 2 imes 3 imes 2 = 60

So, our expression now looks like $60 \sqrt{2}$. We've successfully multiplied all the integers together, and we're left with a single term involving a square root. This is a big step towards our final answer!

Combining the Terms

We've simplified the expression to $60 \sqrt{2}$. Now, let's compare this to the answer choices we were given:

  • A. $15 \sqrt{2}$
  • B. $60 \sqrt{2}$
  • C. $30 \sqrt{3}$
  • D. $60 \sqrt{3}$

Looking at the options, we can see that our simplified expression, $60 \sqrt{2}$, matches option B perfectly. So, the equivalent number to $5 \sqrt{8} \cdot 3 \sqrt{4}$ is $60 \sqrt{2}$.

Double-Checking Our Work

It's always a good idea to double-check our work to make sure we haven't made any mistakes. Let's quickly recap the steps we took:

  1. We identified the expression and the goal (to find the equivalent number).
  2. We simplified the square roots: $\sqrt{8}$ became $2 \sqrt{2}$ and $\sqrt{4}$ became 2.
  3. We substituted the simplified roots back into the original expression.
  4. We multiplied all the integers together.
  5. We compared our simplified expression to the answer choices and found a match.

Each of these steps was crucial to arriving at the correct answer. By breaking down the problem into these smaller parts, we made it much easier to solve. This is a great strategy to use for any math problem that seems overwhelming at first.

Why Option B is Correct

To reiterate, option B, $60 \sqrt{2}$, is the correct answer because it's the simplified form of the original expression $5 \sqrt{8} \cdot 3 \sqrt{4}$. We arrived at this answer by carefully simplifying the square roots and then multiplying all the terms together. The key to success in these types of problems is to be methodical and pay close attention to the details.

Remember, simplifying square roots often involves finding perfect square factors. This is a skill that comes with practice, so the more problems you solve, the better you'll become at it. And don't forget to double-check your work – it can save you from making silly mistakes!

Common Mistakes to Avoid

When working with square roots and radicals, there are a few common mistakes that students often make. Let's go over some of these so you can avoid them in the future:

  1. Forgetting to Simplify: One of the biggest mistakes is not simplifying the square roots as much as possible. Always look for perfect square factors within the square root.
  2. Incorrect Multiplication: Make sure you're multiplying all the terms correctly. It's easy to make a small arithmetic error, especially when there are multiple numbers involved.
  3. Misunderstanding Square Root Properties: Remember the property $\sqrt{a imes b} = \sqrt{a} imes \sqrt{b}$. This is crucial for simplifying square roots, but it only applies to multiplication, not addition or subtraction.
  4. Skipping Steps: It's tempting to try to do everything in your head, but it's much safer to write out each step. This reduces the chance of making a mistake.

By being aware of these common pitfalls, you can increase your chances of getting the correct answer. Math is all about precision, so take your time and be careful with each step.

Practice Makes Perfect

So, there you have it! We've successfully solved the problem and found the equivalent number to $5 \sqrt{8} \cdot 3 \sqrt{4}$. Remember, the key to mastering these types of problems is practice. The more you practice, the more comfortable you'll become with simplifying square roots and performing these calculations.

I encourage you to try some similar problems on your own. Look for questions that involve simplifying radicals and multiplying terms. You can find plenty of resources online or in textbooks. And if you get stuck, don't be afraid to ask for help. There are tons of people who are happy to explain things further.

Further Practice Problems

Here are a couple of practice problems you can try:

  1. Simplify $3 \sqrt{12} imes 2 \sqrt{3}$
  2. What is the equivalent number of $4 \sqrt{18} imes \sqrt{2}$?

Try working through these problems using the same steps we used in this article. Remember to simplify the square roots first, then multiply the terms together. And always double-check your work!

Conclusion

In conclusion, finding the equivalent number of $5 \sqrt{8} \cdot 3 \sqrt{4}$ is a great exercise in simplifying radicals and performing mathematical operations. By breaking down the problem into smaller steps, we were able to arrive at the correct answer, which is $60 \sqrt{2}$. Remember to always simplify square roots, multiply carefully, and double-check your work. And most importantly, keep practicing! You've got this!