Simplifying $4√2(√2+√15)$: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into simplifying a radical expression. We'll break down the steps to multiply and simplify $4√2(√2+√15)$. So, grab your calculators (or don't, we got this!), and let's get started!

Understanding the Problem

Before we jump into the solution, let's clearly understand what we need to do. We're given the expression $4√2(√2+√15)$ and our mission, should we choose to accept it, is to multiply this out and simplify the result as much as possible. This means we need to get rid of any parentheses and combine like terms, all while keeping the expression mathematically sound. Sounds like a plan? Let’s roll!

Why is simplification important, you ask? Well, in mathematics, simplified expressions are like a well-organized closet – easier to work with and understand. A simplified radical expression is one where:

• There are no perfect square factors left under the radical sign.
• There are no radicals in the denominator.
• The fraction inside the radical is in its simplest form.

Simplifying not only makes the expression cleaner but also makes it easier to compare with other expressions and use in further calculations. Think of it as tidying up your mathematical workspace.

Step-by-Step Solution

Step 1: Distribute the $4√2$

The first key step in simplifying this expression involves using the distributive property. Remember this gem from algebra? It states that a(b + c) = ab + ac. We’re going to apply this to our expression. We need to distribute the $4√2$ across both terms inside the parentheses.

So, we multiply $4√2$ by $√2$ and then multiply $4√2$ by $√15$. This gives us:

$4√2 * √2 + 4√2 * √15$

Step 2: Multiply the Radicals

Now, let's focus on those radicals. Remember the rule for multiplying radicals: $√a * √b = √(a * b)$. We'll use this rule to simplify each term.

For the first term, we have $4√2 * √2$. Multiply the numbers under the radical: $√(2 * 2) = √4$. So, the first term becomes $4√4$.

For the second term, we have $4√2 * √15$. Again, multiply the numbers under the radical: $√(2 * 15) = √30$. So, the second term becomes $4√30$.

Putting it together, our expression now looks like this:

$4√4 + 4√30$

Step 3: Simplify the Radicals

Next up, let's simplify those radicals. We need to look for perfect square factors within the radicals. A perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25, etc.).

In the first term, we have $√4$. Guess what? 4 is a perfect square! $√4 = 2$. So, the first term simplifies to $4 * 2 = 8$.

In the second term, we have $√30$. Hmmm, let’s think. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30. None of these (other than 1) are perfect squares. So, $√30$ is already in its simplest form. Bummer! But hey, one down, one to go (or rather, one simplified, one untouched!).

Our expression now looks like this:

$8 + 4√30$

Step 4: Check for Like Terms

Almost there, guys! Now, we need to check if we can combine any like terms. Like terms are terms that have the same radical part. In our expression, we have 8 and $4√30$. 8 is a rational number, and $4√30$ is an irrational number (because of the $√30$). Since they don't have the same radical part (or lack thereof), we can't combine them. They're like apples and oranges – you can't add them directly.

Step 5: Final Answer

And that’s it! We've simplified the expression as much as possible. Our final answer is:

$8 + 4√30$

Common Mistakes to Avoid

Before we wrap up, let's chat about some common pitfalls people often stumble into when simplifying radical expressions. Avoiding these can save you a lot of headaches and ensure you get the correct answer. We've all been there, right? So, let's learn from those mistakes!

• Forgetting to Distribute: A very common mistake is forgetting to distribute the term outside the parentheses to every term inside. Remember, that $4√2$ needs to multiply both the $√2$ and the $√15$. It's like making sure everyone gets a piece of the pie!

• Incorrectly Multiplying Radicals: Remember that $√a * √b = √(a * b)$. You multiply the numbers under the radical, not the numbers outside. For instance, $4√2 * √15$ is $4√(2 * 15)$ which is $4√30$, not $4√(2+15)$.

• Missing Perfect Square Factors: Always, always check if you can simplify the radical further by factoring out perfect squares. If you miss a perfect square, you haven't simplified completely. It's like finding that last hidden sock in the laundry – satisfying to find!

• Combining Unlike Terms: You can only combine terms that have the same radical part. $8 + 4√30$ are not like terms because one is a rational number and the other is an irrational number. It’s tempting to combine them, but resist the urge!

• Incorrectly Simplifying Square Roots: Make sure you know your perfect squares! For example, $√4$ is 2, not 4. It’s a simple mistake, but it can throw off your entire answer. A little review of those squares can go a long way.

Practice Problems

Alright, guys, time to put those newfound skills to the test! Practice makes perfect, as they say. So, here are a few practice problems for you to tackle. Work through them at your own pace, and don't hesitate to revisit the steps we covered if you get stuck.

1. Simplify: $3√5(√5 - √20)$
2. Simplify: $-2√3(√12 + √7)$
3. Simplify: $√2(5√8 - 3√2)$

Remember, the key is to take it step by step, distribute carefully, simplify radicals, and combine like terms. You've got this!

Conclusion

So there you have it! We've successfully multiplied and simplified the expression $4√2(√2+√15)$. We distributed, multiplied radicals, simplified radicals, and identified our final answer: $8 + 4√30$. Remember to avoid those common mistakes, practice regularly, and you'll be simplifying radical expressions like a pro in no time!

Keep up the awesome work, and happy simplifying, mathletes!