How To Simplify The Expression √15(√6 + 6) A Step-by-Step Guide
Introduction
In this article, we will delve into the process of simplifying the expression √15(√6 + 6). This involves applying the principles of algebraic manipulation and the properties of square roots. Our aim is to break down the expression into its simplest form, making it easier to understand and work with. This type of simplification is a fundamental skill in mathematics, often encountered in algebra, calculus, and various other branches. By mastering this technique, you'll enhance your ability to tackle more complex mathematical problems and gain a deeper appreciation for the elegance of mathematical expressions. Understanding how to simplify radical expressions is crucial for success in many areas of mathematics, as it allows us to work with complex equations and formulas more efficiently. Let's embark on this journey of simplifying the given expression, step by step, and uncover the underlying mathematical concepts that make it possible.
Breaking Down the Expression
To simplify the expression √15(√6 + 6), we'll begin by distributing √15 across the terms inside the parentheses. This is a fundamental algebraic technique that allows us to expand the expression and work with each term individually. The distributive property states that a(b + c) = ab + ac, and we'll apply this principle to our expression. This initial step is crucial because it transforms the expression from a product of a radical and a sum into a sum of two terms, each involving a radical. This transformation makes it easier to identify opportunities for further simplification. We'll then focus on simplifying each term separately, using the properties of square roots. This involves breaking down the radicals into their prime factors and looking for pairs of factors that can be extracted from the square root. By carefully applying these techniques, we can systematically reduce the expression to its simplest form. Understanding the distributive property and how it applies to radicals is essential for this process, as it forms the foundation for expanding and simplifying algebraic expressions involving square roots. Let's begin the process by distributing √15 across the terms inside the parentheses.
Step 1: Distribute √15 across the terms inside the parentheses
√15(√6 + 6) = √15 * √6 + √15 * 6
This step involves applying the distributive property, which is a cornerstone of algebra. By distributing √15, we transform the original expression into a sum of two terms, each of which can be simplified independently. This is a crucial step because it allows us to break down the problem into smaller, more manageable parts. The first term, √15 * √6, involves the product of two square roots, which we'll simplify using the property √(a*b) = √a * √b. The second term, √15 * 6, is simply a constant multiplied by a square root, which we'll address later. By carefully applying the distributive property, we set the stage for further simplification and move closer to the final answer. This step highlights the importance of understanding and applying fundamental algebraic principles in simplifying complex expressions. Let's move on to simplifying the individual terms.
Simplifying the Radicals
Now, we focus on simplifying the radicals in the expression √15 * √6 + √15 * 6. The first term involves the product of two square roots, which we can simplify by combining them under a single square root. This is based on the property √(a*b) = √a * √b, which allows us to multiply the radicands (the numbers inside the square roots). The second term involves a square root multiplied by a constant, which we'll handle by leaving the constant outside the radical. Our goal is to break down the radicands into their prime factors and identify any pairs of factors that can be extracted from the square root. This process is crucial for simplifying radicals and expressing them in their simplest form. By carefully applying the properties of square roots and prime factorization, we can reduce the radicals to their simplest forms and make the expression easier to understand. This step demonstrates the power of understanding the properties of radicals and how they can be used to simplify complex expressions. Let's proceed with simplifying the radicals.
Step 2: Simplify √15 * √6
√15 * √6 = √(15 * 6) = √90
Here, we've combined the two square roots into a single square root by multiplying the radicands. This is a direct application of the property √(a*b) = √a * √b. By combining the square roots, we've simplified the expression and made it easier to work with. The next step is to further simplify √90 by breaking down the radicand into its prime factors. This will allow us to identify any pairs of factors that can be extracted from the square root. Understanding this property of square roots is essential for simplifying expressions involving radicals. Let's move on to the next step of breaking down √90.
Step 3: Prime factorize 90
90 = 2 * 45 = 2 * 3 * 15 = 2 * 3 * 3 * 5 = 2 * 3² * 5
Prime factorization is a fundamental technique in number theory and is crucial for simplifying radicals. By breaking down 90 into its prime factors, we can identify any perfect square factors that can be extracted from the square root. In this case, we see that 90 has a factor of 3², which is a perfect square. This means that we can take the square root of 3² and bring it outside the radical. Prime factorization is a powerful tool for simplifying radicals and expressing them in their simplest form. It allows us to identify and extract perfect square factors, making the radical easier to understand and work with. Let's use this prime factorization to further simplify √90.
Step 4: Simplify √90 using the prime factorization
√90 = √(2 * 3² * 5) = √3² * √(2 * 5) = 3√10
In this step, we've used the prime factorization of 90 to simplify the square root. We identified the perfect square factor, 3², and extracted its square root, which is 3. This leaves us with 3√10, which is the simplified form of √90. This process demonstrates the power of prime factorization in simplifying radicals. By identifying and extracting perfect square factors, we can express radicals in their simplest form, making them easier to work with and understand. This skill is essential for simplifying algebraic expressions and solving equations involving radicals. Let's now move on to the next term in the expression.
Step 5: Simplify √15 * 6
√15 * 6 = 6√15
This term is already in a relatively simple form. We have a constant, 6, multiplied by a square root, √15. To see if we can simplify it further, we need to check if √15 can be simplified. We'll do this by breaking down 15 into its prime factors.
Step 6: Prime factorize 15
15 = 3 * 5
The prime factorization of 15 is simply 3 * 5. Since there are no perfect square factors, √15 cannot be simplified further. Therefore, 6√15 remains as it is.
Combining the Simplified Terms
Now that we've simplified both terms, we can combine them to get the simplified expression.
Step 7: Combine the simplified terms
√15 * √6 + √15 * 6 = 3√10 + 6√15
Here, we've simply combined the simplified forms of the two terms we obtained earlier. We have 3√10 and 6√15. Since the radicands (10 and 15) are different, we cannot combine these terms further. They are not like terms, meaning they cannot be added or subtracted directly. Therefore, the expression 3√10 + 6√15 is the simplest form of the original expression. This step highlights the importance of identifying like terms when simplifying algebraic expressions. Only terms with the same radicand can be combined. Let's state our final simplified expression.
Final Simplified Expression
Therefore, the simplified form of the expression √15(√6 + 6) is:
3√10 + 6√15
This is the final answer. We have successfully simplified the expression by distributing, simplifying radicals, and combining like terms. This process demonstrates the importance of understanding the properties of square roots and how to apply them in algebraic manipulation. By breaking down the expression into smaller, more manageable parts, we were able to systematically simplify it to its simplest form. This skill is essential for success in mathematics, as it allows us to work with complex expressions and equations more efficiently. Understanding the steps involved in simplifying radical expressions is crucial for mastering algebra and other advanced mathematical concepts. We have successfully simplified the given expression, and this final result represents the most concise and understandable form of the original expression.
Conclusion
In conclusion, we have successfully simplified the expression √15(√6 + 6) to 3√10 + 6√15. This process involved distributing the √15, simplifying the resulting radicals by breaking them down into their prime factors, and then combining the simplified terms. We utilized key properties of square roots, such as √(a*b) = √a * √b, and the distributive property of multiplication over addition. Understanding these principles is crucial for simplifying algebraic expressions involving radicals. The steps we followed demonstrate a systematic approach to simplifying complex expressions, which can be applied to a wide range of mathematical problems. By breaking down the problem into smaller, more manageable parts, we were able to tackle it effectively and arrive at the final simplified form. This process reinforces the importance of a strong foundation in algebraic manipulation and the properties of radicals. Mastering these skills will not only enhance your ability to solve mathematical problems but also deepen your understanding of mathematical concepts. Simplifying expressions is a fundamental skill in mathematics, and the techniques we've explored here will serve you well in your mathematical journey.