Simplifying Radical Expressions A Step By Step Guide To Solving $3 \sqrt{2}(5 \sqrt{6}-7 \sqrt{3})$

Introduction: Delving into the Realm of Radical Expressions

In the world of mathematics, radical expressions often present a unique challenge, requiring a blend of algebraic manipulation and a solid understanding of the properties of radicals. At the heart of this exploration lies the expression $3 \sqrt{2}(5 \sqrt{6}-7 \sqrt{3})$, a seemingly intricate combination of numbers and square roots. To truly decipher this expression, we must embark on a journey of simplification, employing the fundamental rules of radicals and distribution. This article serves as your comprehensive guide, meticulously breaking down each step to reveal the underlying simplicity and elegance of this mathematical puzzle. We will not only dissect the solution but also delve into the core concepts that empower you to tackle similar radical expressions with confidence.

Dissecting the Expression: A Step-by-Step Simplification

Our quest begins with the expression $3 \sqrt{2}(5 \sqrt{6}-7 \sqrt{3})$. The first key to unlocking its potential lies in the distributive property. This property, a cornerstone of algebra, allows us to multiply the term outside the parentheses, $3 \sqrt{2}$, by each term inside the parentheses. Applying this property, we transform our expression into:

$(3 \sqrt{2} * 5 \sqrt{6}) - (3 \sqrt{2} * 7 \sqrt{3})$

Now, we have two distinct terms to grapple with. Let's focus on the first term, $3 \sqrt{2} * 5 \sqrt{6}$. Here, we can leverage the commutative and associative properties of multiplication to rearrange and regroup the terms:

$(3 * 5) * (\sqrt{2} * \sqrt{6})$

This rearrangement allows us to multiply the coefficients (the numbers outside the square roots) and the radicals (the square roots) separately. Multiplying the coefficients, 3 and 5, gives us 15. Now, let's turn our attention to the radicals. A crucial property of radicals states that the product of two square roots is equal to the square root of the product of the radicands (the numbers inside the square roots). In mathematical notation:

$\sqrt{a} * \sqrt{b} = \sqrt{a * b}$

Applying this property to our expression, we get:

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

Thus, our first term now becomes:

$15 \sqrt{12}$

But we're not quite done yet! The radical $\sqrt{12}$ can be simplified further. To do this, we seek the largest perfect square that divides 12. A perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25). In this case, the largest perfect square that divides 12 is 4 (since 12 = 4 * 3). We can rewrite $\sqrt{12}$ as:

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

Again, using the property $\sqrt{a * b} = \sqrt{a} * \sqrt{b}$, we can separate the radicals:

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

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

$\sqrt{12} = 2 \sqrt{3}$

Substituting this back into our first term, we get:

$15 \sqrt{12} = 15 * 2 \sqrt{3} = 30 \sqrt{3}$

Now, let's tackle the second term in our original expression: $3 \sqrt{2} * 7 \sqrt{3}$. We follow a similar process as before, first regrouping the coefficients and radicals:

$(3 * 7) * (\sqrt{2} * \sqrt{3})$

Multiplying the coefficients, 3 and 7, gives us 21. For the radicals, we again use the property $\sqrt{a} * \sqrt{b} = \sqrt{a * b}$:

$\sqrt{2} * \sqrt{3} = \sqrt{2 * 3} = \sqrt{6}$

Thus, our second term becomes:

$21 \sqrt{6}$

Now, we can substitute the simplified forms of both terms back into our original expression:

$(3 \sqrt{2} * 5 \sqrt{6}) - (3 \sqrt{2} * 7 \sqrt{3}) = 30 \sqrt{3} - 21 \sqrt{6}$

At this point, we need to assess whether we can simplify further. We have two terms, $30 \sqrt{3}$ and $21 \sqrt{6}$. To combine terms with radicals, the radicals must be the same. In this case, we have $\sqrt{3}$ and $\sqrt{6}$, which are different. Moreover, $\sqrt{3}$ is in its simplest form, and $\sqrt{6}$ (which is $\sqrt{2*3}$) cannot be simplified further because neither 2 nor 3 is a perfect square. Therefore, the expression $30 \sqrt{3} - 21 \sqrt{6}$ is in its simplest form.

The Final Result: A Simplified Expression

After meticulously dissecting and simplifying the expression $3 \sqrt{2}(5 \sqrt{6}-7 \sqrt{3})$, we arrive at the final result:

$30 \sqrt{3} - 21 \sqrt{6}$

This expression represents the simplified form of our original expression, showcasing the power of the distributive property and the fundamental rules of radicals. This journey through simplification not only provides the answer but also reinforces the importance of understanding the underlying mathematical principles.

Key Concepts Revisited: Mastering the Art of Radicals

Throughout this exploration, we've encountered several key concepts that are essential for mastering radical expressions. Let's revisit these concepts to solidify our understanding:

• The Distributive Property: This fundamental property allows us to multiply a term by a group of terms within parentheses. In our case, it enabled us to expand the expression and create manageable terms.
• Properties of Radicals: The property $\sqrt{a} * \sqrt{b} = \sqrt{a * b}$ is crucial for simplifying expressions involving the product of radicals. It allows us to combine radicals and then simplify the radicand.
• Simplifying Radicals: The process of finding the largest perfect square that divides the radicand is key to simplifying radicals. This involves factoring the radicand and extracting the square root of the perfect square.
• Combining Like Terms: Radicals can only be combined if they have the same radicand. This is similar to combining like terms in algebraic expressions (e.g., 3x + 2x = 5x).

Beyond the Solution: Applying the Knowledge

The skills and concepts we've explored in this article extend far beyond this specific expression. They form the foundation for tackling a wide range of mathematical problems, including:

• Solving Equations with Radicals: Many equations involve radicals, and the ability to simplify and manipulate them is essential for finding solutions.
• Working with Complex Numbers: Complex numbers often involve square roots of negative numbers, requiring a solid understanding of radical properties.
• Calculus: Radicals appear frequently in calculus, particularly in integration and differentiation.

By mastering these fundamental concepts, you'll be well-equipped to navigate the intricate world of mathematics with confidence and skill. Remember, practice is key! Work through various examples and challenges to further solidify your understanding and unlock the full potential of your mathematical abilities.

Conclusion: The Beauty of Mathematical Simplification

In conclusion, the journey of simplifying $3 \sqrt{2}(5 \sqrt{6}-7 \sqrt{3})$ has been more than just finding an answer. It has been an exploration of fundamental mathematical principles, a testament to the power of simplification, and a journey that has equipped us with valuable tools for tackling future challenges. The final result, $30 \sqrt{3} - 21 \sqrt{6}$, stands as a symbol of the elegance and precision that mathematics offers. Embrace the challenge, delve into the details, and you'll discover the beauty that lies within the world of mathematical expressions.