Simplifying Radicals: A Step-by-Step Guide

Hey guys! Let's break down how to simplify the radical expression $-\sqrt{27} + \sqrt{49} + \sqrt{75}$. Radical expressions might seem intimidating at first, but with a systematic approach, you can simplify them like a pro. We will walk through each term, simplify it, and then combine like terms to get our final answer. This guide is designed to make the process super clear and easy to follow, even if you're just starting with radicals.

Understanding the Basics of Radical Expressions

Before we dive into simplifying the expression, let's cover some basics. A radical expression consists of a radical symbol ($\sqrt{}$), a radicand (the number under the radical), and an index (the small number indicating the root, which is 2 for square roots). Simplifying radical expressions involves breaking down the radicand into its prime factors and pulling out any perfect squares (or perfect cubes, etc., depending on the index). This process makes the expression cleaner and easier to work with.

When you're simplifying, think about finding the largest perfect square that divides evenly into the radicand. For example, if you have $\sqrt{8}$, you can rewrite it as $\sqrt{4 \cdot 2}$, where 4 is a perfect square. This allows you to simplify it to $2\sqrt{2}$. Mastering this technique is key to simplifying more complex expressions. Remember, practice makes perfect, so don't get discouraged if it takes a few tries to get the hang of it. We'll go through plenty of examples to help you become comfortable with this process. Also, keep in mind that understanding prime factorization is super helpful because it lets you systematically break down numbers into their prime factors, making it easier to identify perfect squares. For instance, the prime factorization of 27 is $3 \cdot 3 \cdot 3$, which helps us see that $27 = 3^2 \cdot 3$, making it easy to simplify $\sqrt{27}$. Let's move on to our specific problem now!

Simplifying $-\sqrt{27}$

Let's start by simplifying the first term, $-\sqrt{27}$.

Step 1: Factor the Radicand

We need to find the prime factors of 27. We know that $27 = 3 \times 9$ and $9 = 3 \times 3$. So, we can write 27 as $3 \times 3 \times 3$, or $3^3$.

Step 2: Identify Perfect Squares

Now we look for perfect squares within the factors. We can rewrite $3^3$ as $3^2 \times 3$. Here, $3^2 = 9$ is a perfect square.

Step 3: Rewrite the Radical

Rewrite the radical using the perfect square: $-\sqrt{27} = -\sqrt{3^2 \times 3}$.

Step 4: Simplify

Take the square root of the perfect square and bring it outside the radical: $-\sqrt{3^2 \times 3} = -3\sqrt{3}$.

So, $-\sqrt{27}$ simplifies to $-3\sqrt{3}$. This is our simplified form for the first term. Remember, the negative sign stays outside! Always double-check that you've pulled out the largest possible perfect square to ensure you've fully simplified the expression. Also, keep an eye on the sign, as it's easy to make a mistake and drop it. Now that we've simplified the first term, let's move on to the second term in our original expression.

Simplifying $\sqrt{49}$

Next up, we'll simplify $\sqrt{49}$.

Step 1: Recognize Perfect Square

49 is a perfect square because $7 \times 7 = 49$, which means $49 = 7^2$.

Step 2: Simplify

Taking the square root of 49 is straightforward: $\sqrt{49} = \sqrt{7^2} = 7$.

So, $\sqrt{49}$ simplifies to 7. That was an easy one! Perfect squares are your friends in simplifying radicals, so knowing common ones (like 4, 9, 16, 25, 36, 49, 64, 81, and 100) can save you a lot of time. Make sure you have those memorized, and you'll breeze through these types of problems. Now, let's tackle the last term in our original expression and see what we can do with it.

Simplifying $\sqrt{75}$

Now let's simplify the last term, $\sqrt{75}$.

Step 1: Factor the Radicand

We need to find the prime factors of 75. We know that $75 = 3 \times 25$ and $25 = 5 \times 5$. So, we can write 75 as $3 \times 5 \times 5$, or $3 \times 5^2$.

Step 2: Identify Perfect Squares

We have a perfect square: $5^2 = 25$.

Step 3: Rewrite the Radical

Rewrite the radical using the perfect square: $\sqrt{75} = \sqrt{5^2 \times 3}$.

Step 4: Simplify

Take the square root of the perfect square and bring it outside the radical: $\sqrt{5^2 \times 3} = 5\sqrt{3}$.

So, $\sqrt{75}$ simplifies to $5\sqrt{3}$. Just like with the first term, we identified a perfect square factor and simplified accordingly. Always look for those perfect square factors; they are the key to simplifying these expressions efficiently. Now that we've simplified all three terms, we can put them all together to get our final answer!

Combining the Simplified Terms

Now that we've simplified each term, let's combine them:

$-\sqrt{27} + \sqrt{49} + \sqrt{75} = -3\sqrt{3} + 7 + 5\sqrt{3}$

Step 1: Combine Like Terms

We combine the terms with the same radical, which are $-3\sqrt{3}$ and $5\sqrt{3}$:

$-3\sqrt{3} + 5\sqrt{3} = (-3 + 5)\sqrt{3} = 2\sqrt{3}$

Step 2: Write the Final Expression

Now, we add the constant term 7 to the result:

$2\sqrt{3} + 7$

So, the simplified expression is $7 + 2\sqrt{3}$.

Final Answer

Therefore, $-\sqrt{27} + \sqrt{49} + \sqrt{75} = 7 + 2\sqrt{3}$.

In summary, guys, to simplify radical expressions, break down the radicands into their prime factors, identify and extract perfect squares, and then combine like terms. Practice these steps, and you'll be simplifying radicals with ease. Keep up the great work, and you'll be a radical-simplifying master in no time!