Simplifying $\sqrt{50}+5 \sqrt{72}$ A Step-by-Step Guide
In the realm of mathematics, simplifying radical expressions is a fundamental skill, particularly when dealing with square roots and other radicals. This article delves into the simplification of the expression $\sqrt{50}+5 \sqrt{72}$, providing a step-by-step guide to make the process clear and understandable. We will explore the underlying principles of simplifying radicals, including factoring, identifying perfect squares, and combining like terms. This comprehensive approach will not only assist in solving this specific problem but also equip you with the knowledge to tackle similar mathematical challenges.
Understanding Radicals
Before diving into the simplification process, it's essential to grasp the concept of radicals. A radical, denoted by the symbol √, represents the root of a number. The most common type is the square root, which seeks a number that, when multiplied by itself, equals the number under the radical sign (the radicand). For instance, $\sqrt{9}%$ equals 3 because 3 * 3 = 9. Understanding this basic principle is crucial for simplifying more complex expressions.
Radicals can involve numbers that aren't perfect squares, like $\sqrt{50}$. To simplify these, we look for factors within the radicand that are perfect squares. This process involves breaking down the radicand into its prime factors and identifying pairs. For example, 50 can be factored into 2 * 5 * 5, where 5 * 5 is a perfect square (25). By extracting the square root of the perfect square, we can simplify the radical expression.
Radicals also adhere to certain algebraic rules, such as the product rule, which states that $\sqrt{a * b} = \sqrt{a} * \sqrt{b}$. This rule allows us to separate radicals into simpler forms, making simplification more manageable. In our expression, $\sqrt{50}+5 \sqrt{72}$, we'll use this rule extensively to break down the radicals and identify perfect square factors.
Breaking Down $\sqrt{50}$
Let's begin by focusing on the first term of our expression, $\sqrt{50}$. The key to simplifying this radical lies in finding the prime factors of 50. By breaking down 50, we find that it can be expressed as 2 * 5 * 5. Notice that we have a pair of 5s, which indicates a perfect square factor of 25 (5 * 5).
Now, we can rewrite $\sqrt{50}$ as $\sqrt{25 * 2}$. Applying the product rule of radicals, which states that $\sqrt{a * b} = \sqrt{a} * \sqrt{b}$, we can further separate this into $\sqrt{25} * \sqrt{2}$. Since 25 is a perfect square, its square root is 5. Thus, $\sqrt{25}$ simplifies to 5. The expression now becomes 5 * $\sqrt{2}$, or simply $5\sqrt{2}$. This simplified form is much easier to work with and provides a clearer understanding of the value represented by $\sqrt{50}$.
This process of factoring and extracting perfect squares is fundamental to simplifying any radical expression. By identifying and isolating perfect square factors, we can reduce complex radicals into their simplest forms. In the case of $\sqrt{50}$, we've successfully transformed it into $5\sqrt{2}$, which is its simplest radical form. This step is crucial in our overall goal of simplifying the entire expression, $\sqrt{50}+5 \sqrt{72}$.
Simplifying $5 \sqrt{72}$
Next, we turn our attention to the second term of the expression, $5 \sqrt{72}$. This term presents a slightly more complex radical, but the same principles of factoring and identifying perfect squares apply. Our goal is to simplify $\sqrt{72}$ and then multiply the result by 5.
To begin, we need to find the prime factors of 72. By breaking it down, we find that 72 can be expressed as 2 * 2 * 2 * 3 * 3. Looking at these factors, we can identify two pairs: a pair of 2s and a pair of 3s. This indicates the presence of perfect square factors within 72. Specifically, we have 2 * 2 = 4 and 3 * 3 = 9, both of which are perfect squares.
We can rewrite $\sqrt{72}$ as $\sqrt{4 * 9 * 2}$. Applying the product rule of radicals, we separate this into $\sqrt{4} * \sqrt{9} * \sqrt{2}$. The square roots of 4 and 9 are 2 and 3, respectively. Thus, the expression becomes 2 * 3 * $\sqrt{2}$, which simplifies to $6\sqrt{2}$.
Now, we need to multiply this simplified radical by the coefficient 5. So, $5 \sqrt{72}$ becomes 5 * $6\sqrt{2}$, which equals $30\sqrt{2}$. This simplified form of $5 \sqrt{72}$ is much easier to manage and combine with other terms.
Through this process, we've successfully simplified $5 \sqrt{72}$ to $30\sqrt{2}$. This involved factoring the radicand, identifying perfect square factors, and applying the product rule of radicals. With both terms of our original expression now simplified, we are ready to combine them.
Combining Like Terms
Now that we have simplified both parts of our original expression, $\sqrt50}$ and $5 \sqrt{72}$, we can combine the results. We found that $\sqrt{50}$ simplifies to $5\sqrt{2}$ and $5 \sqrt{72}$ simplifies to $30\sqrt{2}$. Our expression now looks like this + 30\sqrt{2}$.
Combining like terms in radical expressions is similar to combining like terms in algebraic expressions. In this case, both terms have the same radical part, which is $\sqrt{2}$. This means we can add their coefficients. The coefficients are the numbers in front of the radical, which are 5 and 30.
To combine these terms, we simply add the coefficients: 5 + 30 = 35. The radical part, $\sqrt{2}$, remains the same. Therefore, the combined expression is $35\sqrt{2}$. This is the simplest form of our original expression, $\sqrt{50}+5 \sqrt{72}$.
This step of combining like terms is crucial in simplifying radical expressions. It allows us to consolidate multiple terms into a single, simplified form. By identifying terms with the same radical part, we can add or subtract their coefficients to achieve the final simplified expression. In our case, we successfully combined $5\sqrt{2}$ and $30\sqrt{2}$ to arrive at $35\sqrt{2}$, the ultimate simplified form.
Final Simplified Form
After meticulously breaking down and simplifying each term, we have arrived at the final simplified form of the expression $\sqrt{50}+5 \sqrt{72}$. We initially simplified $\sqrt{50}$ to $5\sqrt{2}$ and then simplified $5 \sqrt{72}$ to $30\sqrt{2}$. By combining these like terms, we added their coefficients to obtain the final result.
The final simplified form of the expression is $35\sqrt{2}$. This is the most concise and straightforward representation of the original expression. It showcases the power of simplifying radicals to transform complex expressions into more manageable forms.
This journey through simplifying $\sqrt{50}+5 \sqrt{72}$ highlights the importance of understanding radical properties, prime factorization, and the rules for combining like terms. By applying these principles, we can effectively simplify a wide range of radical expressions, making them easier to work with in various mathematical contexts.
In conclusion, the simplified form of $\sqrt{50}+5 \sqrt{72}$ is $35\sqrt{2}$. This result underscores the elegance and efficiency of mathematical simplification techniques.
Key Takeaways
Simplifying radical expressions is a crucial skill in mathematics. Here are some key takeaways from our exploration of simplifying $\sqrt{50}+5 \sqrt{72}$:
- Prime Factorization: Breaking down radicands into their prime factors is the first step in identifying perfect square factors.
- Product Rule of Radicals: This rule () allows us to separate radicals and simplify them individually.
- Identifying Perfect Squares: Recognizing perfect square factors within the radicand is essential for simplification.
- Combining Like Terms: Terms with the same radical part can be combined by adding or subtracting their coefficients.
- Final Simplified Form: The goal is to express the radical expression in its most concise and manageable form.
By mastering these concepts, you can confidently tackle a variety of radical simplification problems. The process may seem intricate at first, but with practice, it becomes a natural and intuitive part of your mathematical toolkit.
Practice Problems
To further solidify your understanding of simplifying radicals, here are a few practice problems:
- Simplify $\sqrt{18} + 2\sqrt{8}$
- Simplify $3\sqrt{20} - \sqrt{45}$
- Simplify $\sqrt{27} + \sqrt{12}$
Working through these problems will provide you with valuable practice and reinforce the techniques discussed in this article. Remember to focus on prime factorization, identifying perfect squares, and combining like terms. With each problem you solve, your skills in simplifying radicals will grow, enabling you to approach more complex mathematical challenges with confidence.
By mastering the art of simplifying radicals, you unlock a powerful tool for solving a wide array of mathematical problems. This skill is not only essential for academic success but also valuable in various fields that rely on mathematical precision and problem-solving.
