Simplifying The Product Of Radical Expressions A Detailed Solution

Navigating the realm of mathematics, we often encounter expressions that require meticulous simplification. In this article, we delve into the product of two binomials involving square roots: $(2 \sqrt{7}+3 \sqrt{6})(5 \sqrt{2}+4 \sqrt{3})$. Our goal is to unravel this expression, meticulously applying the distributive property and simplifying the resulting terms. By the end of this exploration, we will arrive at the simplified form of the product, shedding light on the underlying mathematical principles at play.

Dissecting the Expression: A Step-by-Step Approach

To embark on this mathematical journey, we will employ the distributive property, a fundamental concept that dictates how to multiply expressions containing multiple terms. The distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last), provides a systematic way to multiply two binomials. In our case, the binomials are $(2 \sqrt{7}+3 \sqrt{6})$ and $(5 \sqrt{2}+4 \sqrt{3})$.

Let's break down the multiplication process step-by-step:

1. First: Multiply the first terms of each binomial: $(2 \sqrt{7}) * (5 \sqrt{2}) = 10 \sqrt{14}$. Here, we multiply the coefficients (2 and 5) and the radicands (7 and 2) separately.
2. Outer: Multiply the outer terms of the binomials: $(2 \sqrt{7}) * (4 \sqrt{3}) = 8 \sqrt{21}$. Again, we multiply the coefficients (2 and 4) and the radicands (7 and 3).
3. Inner: Multiply the inner terms of the binomials: $(3 \sqrt{6}) * (5 \sqrt{2}) = 15 \sqrt{12}$. Notice that the radicand 12 can be further simplified as $12 = 4 * 3$, where 4 is a perfect square.
4. Last: Multiply the last terms of each binomial: $(3 \sqrt{6}) * (4 \sqrt{3}) = 12 \sqrt{18}$. Similarly, the radicand 18 can be simplified as $18 = 9 * 2$, where 9 is a perfect square.

Now, let's combine these results:

$(2 \sqrt{7}+3 \sqrt{6})(5 \sqrt{2}+4 \sqrt{3}) = 10 \sqrt{14} + 8 \sqrt{21} + 15 \sqrt{12} + 12 \sqrt{18}$

Simplifying the Radicals: Unveiling Hidden Patterns

The next crucial step involves simplifying the radicals, identifying and extracting any perfect square factors hidden within the radicands. This simplification process allows us to express the radicals in their simplest form, making it easier to combine like terms.

Let's focus on the radicals that can be simplified:

• $\sqrt{12} = \sqrt{4 * 3} = \sqrt{4} * \sqrt{3} = 2 \sqrt{3}$. We extract the perfect square 4 from the radicand 12.
• $\sqrt{18} = \sqrt{9 * 2} = \sqrt{9} * \sqrt{2} = 3 \sqrt{2}$. Similarly, we extract the perfect square 9 from the radicand 18.

Substituting these simplified radicals back into our expression, we get:

$10 \sqrt{14} + 8 \sqrt{21} + 15(2 \sqrt{3}) + 12(3 \sqrt{2}) = 10 \sqrt{14} + 8 \sqrt{21} + 30 \sqrt{3} + 36 \sqrt{2}$

The Grand Finale: Combining Like Terms

With the radicals simplified, we can now focus on combining like terms. Like terms are those that have the same radical component. In our expression, we have terms with $\sqrt{14}$, $\sqrt{21}$, $\sqrt{3}$, and $\sqrt{2}$.

Examining our expression, we find that there are no other terms with the same radical component as $10 \sqrt{14}$, $8 \sqrt{21}$, $30 \sqrt{3}$, or $36 \sqrt{2}$. Therefore, these terms cannot be combined further.

Thus, the final simplified form of the product is:

$10 \sqrt{14} + 8 \sqrt{21} + 30 \sqrt{3} + 36 \sqrt{2}$

This result matches option D, confirming our solution.

Conclusion: A Triumph of Mathematical Precision

In this article, we embarked on a mathematical journey to unravel the product of $(2 \sqrt{7}+3 \sqrt{6})(5 \sqrt{2}+4 \sqrt{3})$. We meticulously applied the distributive property, simplified radicals, and combined like terms to arrive at the final simplified form: $10 \sqrt{14} + 8 \sqrt{21} + 30 \sqrt{3} + 36 \sqrt{2}$. This exploration highlights the power of mathematical principles in simplifying complex expressions and revealing the underlying beauty of mathematical relationships. The ability to manipulate expressions with radicals is crucial in various fields, from physics and engineering to computer science and cryptography. By mastering these techniques, we empower ourselves to tackle a wide range of mathematical challenges and appreciate the elegance of mathematical solutions.

This detailed, step-by-step explanation not only provides the answer but also reinforces the fundamental concepts involved in simplifying radical expressions. The breakdown of each step, from applying the distributive property to simplifying radicals and combining like terms, ensures a clear understanding of the underlying mathematical principles. This comprehensive approach is essential for building a strong foundation in mathematics and fostering a deeper appreciation for the subject.

Furthermore, the article emphasizes the importance of accuracy and precision in mathematical calculations. Each step is carefully executed and explained, minimizing the possibility of errors. This attention to detail is crucial for achieving correct results and developing problem-solving skills. The process of simplification is not merely about arriving at the answer; it is also about understanding the logical flow of steps and the reasoning behind each manipulation.

Finally, this exploration demonstrates the interconnectedness of mathematical concepts. The distributive property, simplification of radicals, and combining like terms are all essential components of algebraic manipulation. By understanding how these concepts work together, we can approach complex problems with confidence and solve them effectively. The journey of simplifying this particular expression serves as a microcosm of the broader mathematical landscape, illustrating the power of logical reasoning and the beauty of mathematical solutions.