Simplifying Radicals: A Step-by-Step Guide

Welcome, math enthusiasts! Today, we're diving into the fascinating world of radicals and learning how to simplify expressions like $\frac{\sqrt{675}}{\sqrt{3}}$. This process, while seemingly complex at first glance, is actually quite straightforward when broken down into manageable steps. This guide will walk you through the process, ensuring you understand not only how to solve the problem, but also why the steps work.

Understanding the Basics: Radicals and Their Properties

Before we begin, let's establish a solid foundation. The term "radical" refers to the root of a number, often represented by the square root symbol ($\sqrt{ }$). When we see $\sqrt{9}$, for example, we're asking, "What number, when multiplied by itself, equals 9?" The answer, of course, is 3. Similarly, $\sqrt{675}$ asks, "What number, when multiplied by itself, equals 675?" While we could use a calculator to find the approximate answer, our goal here is to simplify the radical expression, not just find a decimal approximation.

Key to simplifying radicals is understanding the properties of radicals, particularly the property that allows us to combine or separate radicals. Specifically, $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$. This property also works in reverse: $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$. This is crucial because it lets us break down a large radical into smaller, more manageable parts. We'll use this extensively as we simplify $\frac{\sqrt{675}}{\sqrt{3}}$. Furthermore, we should also know that $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$. This helps us simplify our original problem by first dividing the numbers inside the radical.

In our case, we're dealing with a fraction of radicals. The ability to manipulate radicals is a fundamental skill in algebra and is essential for solving various mathematical problems. Being able to simplify them means we can find exact solutions and avoid approximations where possible. Moreover, this skillset is valuable in more advanced mathematical fields such as calculus. Therefore, understanding this concept is important in your mathematical journey. Let's start with the first step to calculate the value of $\frac{\sqrt{675}}{\sqrt{3}}$.

Step-by-Step Simplification of $\frac{\sqrt{675}}{\sqrt{3}}$

Now, let's get down to the business of simplifying $\frac{\sqrt{675}}{\sqrt{3}}$. We will begin with the division property of radicals.

Step 1: Combining the Radicals

Using the property $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$, we can rewrite our expression as: $\sqrt{\frac{675}{3}}$. This simplifies the problem into a single radical, making it easier to work with. The expression looks much cleaner now, doesn't it? The beauty of mathematics is how seemingly complex problems can be simplified through the application of the right principles. At this stage, we have only applied the radical rule, without calculating the actual value inside the square root.

Now we proceed to the next step, where we can divide the numbers inside the radical.

Step 2: Performing the Division

Next, we calculate the division inside the radical: $\frac{675}{3} = 225$. So, our expression now becomes $\sqrt{225}$. This step dramatically simplifies our original expression, reducing the size of the number under the radical. It's a great illustration of how the proper use of mathematical rules can make complex problems manageable.

The next step is to find the square root of 225. Let's proceed.

Step 3: Finding the Square Root

Finally, we find the square root of 225. We're looking for a number that, when multiplied by itself, equals 225. In this case, that number is 15, since $15 \times 15 = 225$. Therefore, $\sqrt{225} = 15$. And just like that, we've simplified our original expression to a single whole number!

This simple result shows the power of the methods that we have described above. The ability to simplify radicals is a valuable skill in mathematics, enabling us to efficiently and accurately solve equations, understand geometric concepts, and explore advanced mathematical ideas.

Alternative Approach: Prime Factorization

There's another way to solve this problem, using prime factorization. Prime factorization is the process of breaking down a number into a product of its prime factors. Prime factors are numbers greater than 1 that are only divisible by 1 and themselves. Let's see how this works with our original problem, $\frac{\sqrt{675}}{\sqrt{3}}$.

Step 1: Prime Factorization of 675

First, we find the prime factors of 675. We can start by dividing 675 by the smallest prime number, 3: $675 \div 3 = 225$. Then, we divide 225 by 3 again: $225 \div 3 = 75$. We do this again: $75 \div 3 = 25$. Finally, we divide 25 by 5: $25 \div 5 = 5$. Thus, the prime factorization of 675 is $3 \times 3 \times 3 \times 5 \times 5$ or $3^3 \times 5^2$.

This is a fundamental skill in number theory, helping us to understand the structure of numbers and simplifying complex calculations. Prime factorization reveals the building blocks of a number. Now, how do we use this with the original problem?

Step 2: Prime Factorization of 3

The prime factorization of 3 is simply 3, as 3 is itself a prime number. We can then rewrite our original expression as $\frac{\sqrt{3^3 \times 5^2}}{\sqrt{3}}$. Now we apply the rule of division of the radicals to have $\sqrt{\frac{3^3 \times 5^2}{3}}$. Notice that the prime factorizations are handy because they break down the numbers into their fundamental components, making it easier to simplify and work with them.

Step 3: Simplifying the Expression Using Prime Factors

Now, let's simplify our new expression $\sqrt{\frac{3^3 \times 5^2}{3}}$. First, we divide the powers of 3. We have $3^3$ in the numerator and $3^1$ in the denominator, so $3^3 \div 3^1 = 3^2$. The expression simplifies to $\sqrt{3^2 \times 5^2}$. Then, we apply the property $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$. Our expression can be rewritten as $\sqrt{3^2} \times \sqrt{5^2} = 3 \times 5 = 15$. And again, we arrive at the answer 15! This approach highlights the interconnectedness of different mathematical concepts and the power of breaking down a problem into its fundamental parts. In summary, we have the same result using the prime factorization method.

Conclusion: Mastering Radical Simplification

We've successfully simplified $\frac{\sqrt{675}}{\sqrt{3}}$ using two different methods: combining the radicals and directly calculating the division, and using prime factorization. Both methods lead to the same answer, 15, reinforcing the importance of understanding the properties of radicals and how to manipulate them. By breaking down the problem into smaller steps and understanding the underlying principles, what seemed complex at first becomes manageable.

The ability to simplify radicals is a fundamental skill in mathematics. Whether you're working on algebra problems, tackling geometry questions, or exploring calculus concepts, knowing how to manipulate radicals will serve you well. Keep practicing, and you'll become more and more comfortable with this important skill. Remember to always double-check your work and to understand why each step is necessary. With practice and persistence, you'll be simplifying radicals with ease!

For further practice and more examples, check out resources on Khan Academy's Radical Expressions. This website provides comprehensive explanations and practice exercises.