Simplify Radical Expression 5√2 ⋅ 9√6 A Step By Step Solution

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This comprehensive guide will walk you through simplifying the radical expression $5\sqrt{2} \cdot 9\sqrt{6}$. Radical expressions, which involve square roots or other roots, often appear complex, but with a systematic approach, they can be simplified to a more manageable form. This article will break down each step, ensuring you understand the underlying principles and can apply them to other similar problems. We'll explore the properties of radicals, demonstrate the multiplication process, and guide you to the final simplified answer. Whether you're a student tackling algebra or just looking to refresh your math skills, this guide will provide the clarity and practice you need to master simplifying radical expressions.

Understanding the Basics of Radical Expressions

To effectively simplify the expression $5\sqrt{2} \cdot 9\sqrt{6}$, it's crucial to understand the fundamental properties of radicals. A radical expression consists of a radical symbol (√), a radicand (the number under the radical), and an index (the degree of the root). In the case of square roots, the index is 2, though it's often omitted. For instance, in $\sqrt{2}$, the radical symbol is √, the radicand is 2, and the index is implicitly 2. Understanding these components is the first step in manipulating radical expressions correctly.

One of the key properties we'll use is the product rule for radicals, which states that the square root of a product is the product of the square roots: $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$. This rule allows us to break down complex radicals into simpler forms. Additionally, we'll use the property that the product of radicals can be simplified if the radicands have common factors that are perfect squares. Recognizing perfect squares (e.g., 4, 9, 16, 25) is crucial for simplifying radicals effectively. For example, $\sqrt{16}$ simplifies to 4 because 16 is a perfect square.

Another important concept is the simplification of radicals by factoring out perfect squares. If the radicand has factors that are perfect squares, we can take the square root of those factors and move them outside the radical. For example, to simplify $\sqrt{8}$, we can rewrite it as $\sqrt{4 \cdot 2}$, which then simplifies to $\sqrt{4} \cdot \sqrt{2}$, and finally to $2\sqrt{2}$. This process involves identifying perfect square factors within the radicand and applying the product rule in reverse. Mastering these basics will make the simplification process smoother and more intuitive. Furthermore, understanding how to combine and multiply radicals is essential. We treat the coefficients (numbers outside the radical) and the radicals separately. For example, in the expression $5\sqrt{2}$, 5 is the coefficient and $\sqrt{2}$ is the radical part. When multiplying radical expressions, we multiply the coefficients together and the radicals together, adhering to the product rule mentioned earlier. By grasping these foundational concepts, you'll be well-prepared to tackle the simplification of $5\sqrt{2} \cdot 9\sqrt{6}$ and other similar expressions.

Step-by-Step Simplification of $5\sqrt{2} \cdot 9\sqrt{6}$

Now, let's dive into the step-by-step simplification of the given expression, $5\sqrt{2} \cdot 9\sqrt{6}$. This process involves multiplying the coefficients and the radicals separately, and then simplifying the resulting radical expression.

Step 1: Multiply the Coefficients

The first step in simplifying $5\sqrt{2} \cdot 9\sqrt{6}$ is to multiply the coefficients. The coefficients are the numbers outside the radical, which in this case are 5 and 9. Multiplying these gives us:

$5 \cdot 9 = 45$

So, the coefficient of our simplified expression will be 45. This initial step is straightforward but crucial as it sets the foundation for the rest of the simplification process. Correctly multiplying the coefficients is essential for arriving at the correct final answer.

Step 2: Multiply the Radicals

The next step is to multiply the radicals. We have $\sqrt{2}$ and $\sqrt{6}$. According to the product rule for radicals, we can multiply these by combining them under a single radical:

$\sqrt{2} \cdot \sqrt{6} = \sqrt{2 \cdot 6} = \sqrt{12}$

Now we have $\sqrt{12}$. This simplifies the expression to $45\sqrt{12}$. However, we are not done yet, as $\sqrt{12}$ can be further simplified. Multiplying the radicals together is a key step, but it's equally important to recognize when the resulting radical can be simplified further.

Step 3: Simplify the Resulting Radical

The radical $\sqrt{12}$ can be simplified by identifying perfect square factors within the radicand (12). We can rewrite 12 as a product of its factors:

$12 = 4 \cdot 3$

Notice that 4 is a perfect square ($2^2$). Thus, we can rewrite $\sqrt{12}$ as:

$\sqrt{12} = \sqrt{4 \cdot 3}$

Using the product rule for radicals in reverse, we can separate this into:

$\sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3}$

Since $\sqrt{4} = 2$, we have:

$2\sqrt{3}$

Simplifying the radical involves factoring out perfect squares and reducing the expression to its simplest form.

Step 4: Combine the Simplified Radical with the Coefficient

Now we combine the simplified radical $2\sqrt{3}$ with the coefficient we calculated earlier (45):

$45 \cdot 2\sqrt{3} = 45 \cdot 2 \cdot \sqrt{3} = 90\sqrt{3}$

So, the simplified form of $5\sqrt{2} \cdot 9\sqrt{6}$ is $90\sqrt{3}$. This final step brings together all the previous simplifications, resulting in the most reduced form of the expression. Combining the simplified radical with the coefficient is the ultimate step in achieving the solution.

Detailed Explanation of the Solution

To recap, let's walk through the detailed explanation of how we arrived at the simplified form of $5\sqrt{2} \cdot 9\sqrt{6}$. Understanding the rationale behind each step is crucial for grasping the broader concepts of simplifying radical expressions.

Initial Expression: $5\sqrt{2} \cdot 9\sqrt{6}$

We start with the given expression, $5\sqrt{2} \cdot 9\sqrt{6}$. Our goal is to simplify this expression by multiplying the coefficients and the radicals, and then reducing the resulting radical to its simplest form. The initial expression sets the stage for the simplification process.

Step 1: Multiplying Coefficients

The coefficients are the numbers outside the radical symbols, which are 5 and 9. Multiplying these together gives us:

$5 \cdot 9 = 45$

This means that the numerical part of our simplified expression will be 45. This step is a straightforward application of multiplication but is essential for the overall simplification. Multiplying the coefficients provides the numerical component of our final answer.

Step 2: Multiplying Radicals

Next, we multiply the radicals, $\sqrt{2}$ and $\sqrt{6}$. According to the product rule for radicals, we can combine these under a single radical:

$\sqrt{2} \cdot \sqrt{6} = \sqrt{2 \cdot 6} = \sqrt{12}$

This simplifies the radical part of our expression to $\sqrt{12}$. Now, our expression looks like $45\sqrt{12}$. Multiplying the radicals is a key step, but it often leads to a radical that can be further simplified.

Step 3: Simplifying the Radical $\sqrt{12}$

To simplify $\sqrt{12}$, we look for perfect square factors within the radicand (12). The factors of 12 are 1, 2, 3, 4, 6, and 12. Among these, 4 is a perfect square ($2^2$). We can rewrite 12 as:

$12 = 4 \cdot 3$

Thus, we can rewrite $\sqrt{12}$ as:

$\sqrt{12} = \sqrt{4 \cdot 3}$

Using the product rule for radicals, we separate this into:

$\sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3}$

Since $\sqrt{4} = 2$, we have:

$2\sqrt{3}$

So, the simplified form of $\sqrt{12}$ is $2\sqrt{3}$. Simplifying the radical involves identifying and factoring out perfect squares, which is a critical step in reducing the expression to its simplest form.

Step 4: Combining the Simplified Radical with the Coefficient

Finally, we combine the simplified radical $2\sqrt{3}$ with the coefficient 45:

$45 \cdot 2\sqrt{3} = 45 \cdot 2 \cdot \sqrt{3} = 90\sqrt{3}$

Therefore, the simplified form of the original expression $5\sqrt{2} \cdot 9\sqrt{6}$ is $90\sqrt{3}$. Combining the simplified radical with the coefficient is the final step that yields the most reduced form of the expression. This thorough explanation underscores the importance of each step in the simplification process, from multiplying coefficients and radicals to identifying and factoring out perfect squares. By understanding these steps, you can confidently simplify a wide range of radical expressions.

Conclusion: The Simplified Form

In conclusion, after carefully following each step of the simplification process, we have determined that the simplified form of the expression $5\sqrt{2} \cdot 9\sqrt{6}$ is $90\sqrt{3}$. This result is achieved by first multiplying the coefficients (5 and 9) to get 45, then multiplying the radicals ($\sqrt{2}$ and $\sqrt{6}$) to get $\sqrt{12}$. The crucial step is simplifying $\sqrt{12}$ by recognizing that it can be factored into $\sqrt{4 \cdot 3}$, which simplifies to $2\sqrt{3}$. Finally, multiplying the simplified radical by the coefficient (45) gives us the final answer, $90\sqrt{3}$.

Understanding how to simplify radical expressions is a fundamental skill in algebra. It involves the application of the product rule for radicals, the identification of perfect square factors, and the methodical combination of coefficients and radicals. This step-by-step guide has demonstrated how to break down a seemingly complex expression into manageable parts, ensuring that each operation is performed correctly to arrive at the simplified form. The ability to simplify radicals is not only useful for academic purposes but also for various real-world applications where mathematical expressions need to be presented in their simplest form.

By mastering these techniques, you'll be well-equipped to tackle more advanced mathematical problems involving radicals. The key is to practice and become comfortable with each step, from identifying coefficients and radicals to factoring and simplifying. Remember, the goal is to present the expression in its most reduced form, where the radicand has no perfect square factors other than 1. With consistent effort and a clear understanding of the principles involved, simplifying radical expressions will become a routine task. This skill is an invaluable asset in your mathematical toolkit, empowering you to approach and solve a wide range of problems with confidence.

Therefore, the correct answer is D. $90\sqrt{3}$.