Simplifying $2√5(3√2+√45-4√5)$: A Math Guide

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Let's dive into the world of radicals and simplify the expression $2√5(3√2+√45-4√5)$. This article will break down each step, making it easy to understand even if you're not a math whiz. We’ll cover the basic principles of simplifying radicals and then apply them to the given expression. By the end of this guide, you'll not only know how to solve this particular problem but also have a better understanding of how to tackle similar mathematical challenges. So, let's get started and make math a little less daunting!

Understanding the Basics of Radicals

Before we jump into the main problem, let's quickly recap what radicals are and how we can simplify them. In mathematical terms, a radical is an expression that uses a root, such as a square root, cube root, etc. The most common type is the square root, denoted by the symbol √. When you see √a, it means “the square root of a,” which is the number that, when multiplied by itself, equals a.

Simplifying radicals often involves breaking down the number inside the square root into its prime factors. The main goal here is to identify any perfect square factors because the square root of a perfect square is an integer. For example, consider √16. Since 16 is 4 * 4 (or 4²), it’s a perfect square, and √16 simplifies to 4. Let's take another example: √20. We can rewrite 20 as 4 * 5, where 4 is a perfect square. Thus, √20 becomes √(4 * 5), which can be further simplified as √4 * √5 = 2√5. This method is crucial for making radical expressions easier to work with.

Key properties that help us simplify radicals include:

• √(a * b) = √a * √b (The square root of a product is the product of the square roots)
• √(a²) = a (The square root of a square is the number itself)

These properties are not just abstract rules; they are the tools we use to dissect and simplify complex radical expressions. Understanding and applying these properties will make working with radicals much more manageable and less intimidating. Keep these principles in mind as we move forward to tackle our main problem.

Step-by-Step Simplification of $2√5(3√2+√45-4√5)$

Now, let’s tackle the expression $2√5(3√2+√45-4√5)$ step by step. We'll start by distributing the $2√5$ across the terms inside the parentheses. This is similar to how you would handle any algebraic expression where you need to multiply a term by a group of terms. By carefully distributing and simplifying each term, we’ll break down the problem into smaller, more manageable parts.

Step 1: Distribute $2√5$

The first step involves multiplying $2√5$ by each term inside the parentheses:

$2√5 * (3√2) + 2√5 * (√45) - 2√5 * (4√5)$

This gives us three separate terms to simplify:

$6√10 + 2√(5 * 45) - 8√(5 * 5)$

Distributing the term is a fundamental algebraic technique that helps us unravel complex expressions. It allows us to address each part individually before combining them back together. This approach not only simplifies the immediate step but also sets the stage for easier simplification in subsequent steps. Make sure you take your time and double-check your distribution to avoid errors early on.

Step 2: Simplify Each Term

Now, let’s simplify each term individually. This involves looking for perfect squares within the radicals and simplifying them. We’ll focus on breaking down the numbers inside the square roots into their prime factors, making it easier to identify and extract perfect squares. This step is crucial for reducing the expression to its simplest form.

Term 1: $6√10$

$6√10$ can be written as $6√(2 * 5)$. Since there are no perfect square factors, this term remains as $6√10$.

Term 2: $2√(5 * 45)$

Let's simplify $2√(5 * 45)$. First, multiply 5 and 45 to get:

$2√225$

Now, we recognize that 225 is a perfect square ($15^2 = 225$), so:

$2√225 = 2 * 15 = 30$

Term 3: $8√(5 * 5)$

Next, we simplify $8√(5 * 5)$. We see that:

$8√(5 * 5) = 8√25$

Since 25 is a perfect square ($5^2 = 25$), we have:

$8√25 = 8 * 5 = 40$

By simplifying each term individually, we’ve transformed the expression into a set of much simpler components. This methodical approach is key to solving more complex problems. Each simplification brings us closer to the final answer, making the overall process more manageable.

Step 3: Combine the Simplified Terms

Now that we've simplified each term, let's combine them. Our expression has been reduced to:

$6√10 + 30 - 40$

Combine the constants:

$6√10 - 10$

This is the simplest form of the expression.

By combining the simplified terms, we arrive at the final form of our expression. This step demonstrates how the individual pieces fit together to create the complete solution. Make sure to double-check your arithmetic during this phase to avoid any minor errors that can affect the final result. The ability to combine terms correctly is a cornerstone of algebraic manipulation.

Final Result and Key Takeaways

Therefore, the simplified form of the expression $2√5(3√2+√45-4√5)$ is $6√10 - 10$. We successfully navigated through the problem by breaking it down into manageable steps: distributing, simplifying individual terms, and then combining like terms. Each step built upon the previous one, leading us to the final solution. This methodical approach is not just useful for simplifying radicals but can be applied to a wide range of mathematical problems.

The key takeaways from this exercise are:

1. Distribute First: When you have a term multiplying a parenthesis, always distribute it across each term inside.
2. Simplify Radicals: Look for perfect square factors within the radicals and simplify them.
3. Combine Like Terms: Combine constants and any like radical terms to reach the simplest form.

By understanding and applying these principles, you can confidently tackle similar problems. Remember, practice makes perfect, so keep working on different types of expressions to build your skills. Simplifying radicals can seem challenging at first, but with a clear understanding of the rules and a systematic approach, you can master these types of problems.

Practice Problems

To further solidify your understanding, let’s look at a couple of practice problems. Working through these will help you apply the techniques we've discussed and build confidence in your ability to simplify radical expressions. Practice is an essential part of learning mathematics, and these exercises are designed to reinforce the steps and concepts we’ve covered. Let's put your new skills to the test!

Practice Problem 1

Simplify the expression: $3√2(2√8 - √18 + 5√2)$

Practice Problem 2

Simplify: $√3(4√12 + 2√27 - √3)$

Try solving these on your own, referring back to the steps we discussed earlier if needed. The answers will be provided below, but make sure to attempt the solutions yourself first. Working through the process is where the real learning happens. Solving these problems will not only improve your understanding but also make you more comfortable with simplifying radical expressions.

Answers:

1. $3√2(2√8 - √18 + 5√2) = 24$
2. $√3(4√12 + 2√27 - √3) = 21$

Conclusion

Simplifying radical expressions might seem daunting initially, but by breaking down the problem into manageable steps and consistently applying the rules, it becomes much more approachable. We started by understanding the basics of radicals, then moved on to simplifying the given expression step by step. We distributed, simplified individual terms, combined like terms, and finally arrived at the simplest form. Through practice problems, we reinforced these techniques and built confidence in tackling similar mathematical challenges.

Remember, the key to mastering math is practice and a systematic approach. Each problem you solve adds to your understanding and skills, making you a more confident and capable mathematician. So, keep practicing, keep learning, and don’t hesitate to revisit the concepts whenever needed. With dedication and the right strategies, you can conquer any mathematical problem that comes your way.

For further learning and to explore more resources on simplifying radical expressions, visit trusted websites like Khan Academy's Radicals Section.