Multiplying And Simplifying Radicals $6 \sqrt{6}(4+5 \sqrt{14})$

Have you ever stumbled upon a math problem that looks like a tangled mess of square roots and wondered where to even begin? Well, guys, you're not alone! Multiplying radicals can seem intimidating at first, but with a little know-how and some practice, you'll be simplifying these expressions like a pro. In this article, we're going to break down the process step-by-step, focusing on a specific example to illustrate the key concepts. So, buckle up and let's dive into the world of radical multiplication!

Understanding Radicals

Before we jump into the multiplication, let's make sure we're all on the same page about what radicals actually are. A radical is simply a root of a number. The most common radical is the square root, denoted by the symbol √. The square root of a number x is the value that, when multiplied by itself, equals x. For example, the square root of 9 is 3 because 3 * 3 = 9. The number inside the radical symbol is called the radicand. Radicals can also represent cube roots, fourth roots, and so on. The nth root of a number x is written as ⁿ√x, where n is the index of the radical. In the case of a square root, the index is implicitly 2.

Radical expressions often involve a coefficient, which is the number multiplied by the radical. For instance, in the expression 6√6, 6 is the coefficient and 6 is the radicand. When multiplying radical expressions, we'll need to handle both the coefficients and the radicands separately.

Key Properties of Radicals

To effectively multiply and simplify radicals, it's crucial to understand a few key properties:

1. Product Property: The square root of a product is equal to the product of the square roots. Mathematically, this is expressed as √(a * b) = √a * √b. This property allows us to break down complex radicals into simpler forms.
2. Simplifying Radicals: A radical is in its simplest form when the radicand has no perfect square factors (other than 1). To simplify a radical, we look for the largest perfect square that divides the radicand and then use the product property to extract the square root of that factor.

With these basics in mind, we're ready to tackle our main problem: multiplying and simplifying the expression 6√6(4 + 5√14).

Multiplying the Radicals: A Step-by-Step Approach

Our mission, should we choose to accept it, is to multiply and simplify the expression 6√6(4 + 5√14). This problem involves multiplying a radical expression by a binomial containing another radical. Don't worry, guys, it's not as scary as it looks! We'll use the distributive property, just like we would with any algebraic expression.

The distributive property states that a(b + c) = ab + ac. In our case, a = 6√6, b = 4, and c = 5√14. So, we need to distribute 6√6 to both terms inside the parentheses:

6√6(4 + 5√14) = (6√6 * 4) + (6√6 * 5√14)

Now, let's tackle each term separately:

Term 1: 6√6 * 4

This is pretty straightforward. We simply multiply the coefficients:

6√6 * 4 = 6 * 4 * √6 = 24√6

So, the first term is 24√6. Easy peasy!

Term 2: 6√6 * 5√14

This term requires a little more attention. We need to multiply both the coefficients and the radicands:

6√6 * 5√14 = (6 * 5) * (√6 * √14) = 30√(6 * 14)

Now, let's multiply the numbers inside the radical:

30√(6 * 14) = 30√84

So, the second term is 30√84. But we're not done yet! We need to simplify this radical.

Simplifying the Resulting Radicals

We've multiplied the radicals, but to give our answer in the simplest form, we need to check if we can simplify the radicals further. Remember, a radical is in its simplest form when the radicand has no perfect square factors (other than 1).

Let's look at our two terms:

1. 24√6: The radicand is 6. The factors of 6 are 1, 2, 3, and 6. None of these (except 1) are perfect squares, so √6 is already in its simplest form.
2. 30√84: The radicand is 84. Let's find the prime factorization of 84: 84 = 2 * 2 * 3 * 7 = 2² * 3 * 7. Aha! We see a perfect square factor: 2² = 4. So, we can simplify √84.

Simplifying √84

Using the product property of radicals, we can write:

√84 = √(4 * 21) = √4 * √21 = 2√21

Now, let's substitute this back into our second term:

30√84 = 30 * 2√21 = 60√21

Putting It All Together: The Final Simplified Answer

We've done the hard work! Now, let's combine our simplified terms:

6√6(4 + 5√14) = 24√6 + 60√21

Can we simplify this expression further? Well, we need to check if the radicals have any common factors. The radicands are 6 and 21. The factors of 6 are 1, 2, 3, and 6. The factors of 21 are 1, 3, 7, and 21. The greatest common factor is 3, but neither 6 nor 21 is divisible by a perfect square greater than 1. Therefore, we cannot simplify the expression any further.

So, our final answer, in simplest form, is:

24√6 + 60√21

Key Takeaways and Practice Makes Perfect

Guys, we've successfully multiplied and simplified a radical expression! Let's recap the key steps:

1. Distribute: Use the distributive property to multiply the radical expression by each term inside the parentheses.
2. Multiply Coefficients and Radicands: Multiply the coefficients together and the radicands together.
3. Simplify Radicals: Look for perfect square factors in the radicands and use the product property to simplify.
4. Combine Like Terms: If possible, combine terms with the same radicand.

Multiplying and simplifying radicals is a fundamental skill in algebra. The more you practice, the more comfortable you'll become with the process. Try tackling different problems with varying levels of complexity. Remember to always look for opportunities to simplify the radicals to express your answer in its simplest form.

So, there you have it! You've now got the tools and knowledge to multiply and simplify radical expressions like a champ. Keep practicing, and you'll be a radical master in no time!