Solving √5 × √10 - √8: A Step-by-Step Guide

Hey guys! Today, we're diving into a fun math problem that involves square roots. We're going to break down how to solve √5 × √10 - √8 step by step. This is a common type of question you might see in math class or on a test, so let's get started and make sure you understand it completely!

Understanding the Problem

Before we jump into solving, let's make sure we really understand what the question is asking. We need to find the value of the expression √5 × √10 - √8. This involves multiplying square roots and then subtracting another square root. Sounds like a plan? Let's break it down bit by bit.

Breaking Down the Components

First, let’s identify the key parts of the expression:

• √5: This is the square root of 5. We can't simplify it further as 5 is a prime number.
• √10: This is the square root of 10. We can think of 10 as 5 × 2, so √10 can be seen as √(5 × 2).
• √8: This is the square root of 8. We can break 8 down into 2 × 2 × 2, or 2³. This will be important when we simplify.

Why This Matters

Understanding these components is crucial because it allows us to simplify the expression. Remember, the goal in math is often to make things as simple as possible while keeping the value the same. Simplifying square roots involves finding perfect squares within the roots.

Now that we've identified the pieces, let's move on to the step-by-step solution. We’ll take it nice and slow, so you can follow along easily. Ready? Let's do this!

Step 1: Multiply √5 and √10

Okay, so the first part of our problem is √5 × √10. When we multiply square roots, we can use a handy rule: √(a) × √(b) = √(a × b). This means we can multiply the numbers inside the square roots together.

Applying the Rule

Let’s use this rule for our problem:

√5 × √10 = √(5 × 10) = √50

So, we've turned √5 × √10 into √50. But we're not done yet! We need to see if we can simplify √50 further.

Simplifying √50

To simplify √50, we need to find the largest perfect square that divides 50. A perfect square is a number that can be obtained by squaring an integer (like 4, 9, 16, 25, etc.).

Think about the factors of 50: 1, 2, 5, 10, 25, and 50. Among these, 25 is a perfect square (since 5² = 25). So, we can rewrite 50 as 25 × 2.

Now, let's rewrite √50:

√50 = √(25 × 2)

Using our rule from before, we can separate this into:

√(25 × 2) = √25 × √2

We know that √25 = 5, so we get:

5√2

Woo-hoo! We’ve simplified √5 × √10 to 5√2. Now, let's move on to the next part of the problem.

Step 2: Simplify √8

Next up, we need to simplify √8. Just like we did with √50, we want to find the largest perfect square that divides 8. Think about the factors of 8: 1, 2, 4, and 8. We see that 4 is a perfect square (since 2² = 4).

Breaking Down √8

We can rewrite 8 as 4 × 2. So, √8 becomes:

√8 = √(4 × 2)

Now, we use our trusty rule again to separate the square root:

√(4 × 2) = √4 × √2

We know that √4 = 2, so we have:

2√2

Awesome! We've simplified √8 to 2√2. Now we're ready to put it all together.

Why Simplifying Matters

Simplifying square roots makes them easier to work with. It's like tidying up before you start a big project – it makes everything flow smoother. By simplifying √50 and √8, we've made the final calculation much easier.

Step 3: Subtract the Simplified Roots

Alright, we've done the hard work of simplifying the square roots. Now we just need to subtract them. Remember, our expression is √5 × √10 - √8, which we've simplified to 5√2 - 2√2.

Subtracting Like Terms

Think of √2 as a variable, like 'x'. So, we have 5√2 - 2√2, which is similar to 5x - 2x. We can combine these because they have the same square root part (√2).

To subtract, we simply subtract the numbers in front of the square roots:

5√2 - 2√2 = (5 - 2)√2

So, 5 - 2 = 3, and our expression becomes:

3√2

And there you have it! We've solved the problem. √5 × √10 - √8 simplifies to 3√2.

Final Answer and Multiple Choices

So, the final answer to our problem is 3√2. Now, let’s take a look at the multiple-choice options you provided:

• a. 2√2
• b. 3√2
• c. 4√2
• d. 5√2
• e. 6√2

We can clearly see that option b. 3√2 is the correct answer.

Double-Checking Your Work

It's always a good idea to double-check your work, especially in math. Make sure you’ve followed each step correctly and haven't made any simple errors. In this case, we've carefully simplified each square root and then subtracted them correctly.

Why This Problem Matters

Understanding how to work with square roots is super important in math. It's not just about getting the right answer on a test. It's about building a solid foundation for more advanced topics in algebra, geometry, and beyond. Plus, it's kinda cool to see how these numbers work together, right?

Practical Applications

Square roots pop up in all sorts of real-world situations. From calculating distances using the Pythagorean theorem to understanding areas and volumes, they’re everywhere. So, mastering these skills now will definitely pay off in the long run.

Tips for Mastering Square Roots

Okay, guys, let’s talk about how you can really nail working with square roots. Here are a few tips to help you become a square root superstar!

1. Practice Regularly

The key to getting better at anything in math is practice. The more you work with square roots, the more comfortable you'll become. Try doing a few problems every day to keep the concepts fresh in your mind.

2. Know Your Perfect Squares

Memorizing the perfect squares (1, 4, 9, 16, 25, 36, etc.) can make simplifying square roots much faster. When you see a number like √36, you'll immediately know it's 6.

3. Break It Down

When you encounter a larger number under the square root, break it down into its factors. Look for perfect squares within those factors. This is exactly what we did when we simplified √50 and √8.

4. Use the Rules

Remember the rules for multiplying and dividing square roots: √(a) × √(b) = √(a × b) and √(a) / √(b) = √(a / b). These rules are your best friends when working with square roots.

5. Check Your Work

Always double-check your work. Make sure you haven’t made any simple mistakes, especially when subtracting or adding. It’s easy to slip up, so take a moment to review each step.

6. Visualize It

Sometimes, visualizing square roots can help. Think about a square with an area of 25 square units. The side length would be √25, which is 5 units. This can give you a more intuitive understanding of what a square root represents.

Conclusion

So, guys, we've successfully solved the problem √5 × √10 - √8, and we found the answer to be 3√2. We broke it down step by step, from multiplying the square roots to simplifying and finally subtracting. Remember, the key to mastering these problems is understanding the rules and practicing regularly. You've got this!

Keep practicing, keep exploring, and most importantly, keep having fun with math. You’ll be surprised at how much you can achieve. Until next time, happy calculating!