Simplifying $3√7 + 9√175$: A Step-by-Step Guide

Nov 30, 2025 by Alex Johnson 48 views

In this comprehensive guide, we will walk through the process of simplifying the expression $3√7 + 9√175$ . This type of problem often appears in algebra and requires a good understanding of radicals and their properties. By the end of this article, you'll not only know how to simplify this specific expression but also gain valuable skills for tackling similar mathematical challenges. Let’s dive in!

Understanding the Basics of Radicals

Before we jump into the simplification, let's quickly review what radicals are and how they work. A radical, often denoted by the symbol √, represents a root of a number. For example, √9 is the square root of 9, which is 3 because 3 * 3 = 9. Similarly, √25 is 5 because 5 * 5 = 25. When dealing with expressions involving radicals, it’s essential to identify perfect squares (or perfect cubes, etc., depending on the root) within the radical.

In our expression, $3√7 + 9√175$ , we have two terms involving square roots: $√7$ and $√175$ . The key to simplifying this expression lies in breaking down the radicals to their simplest forms. Specifically, we want to see if we can find any perfect square factors within the radicand (the number under the radical sign). This is a fundamental concept in simplifying radical expressions.

Radicals are a cornerstone of algebra and calculus, making it critical to understand how to manipulate them effectively. The rules of radicals allow us to combine, divide, multiply, and simplify these expressions, and they play a crucial role in various mathematical contexts. This detailed guide will give you a strong foundation in working with radicals.

Step 1: Simplify √175

The first step in simplifying $3√7 + 9√175$ is to break down $√175$ . To do this, we need to find the largest perfect square that divides 175. A perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25, etc.).

Let's consider the factors of 175. We can start by dividing 175 by small perfect squares:

175 ÷ 4 = 43.75 (not an integer)
175 ÷ 9 = 19.44 (not an integer)
175 ÷ 16 = 10.9375 (not an integer)
175 ÷ 25 = 7 (an integer!)

We found that 175 can be divided by 25, which is a perfect square (5 * 5 = 25). So, we can rewrite 175 as 25 * 7. Now, we can rewrite $√175$ as $√(25 * 7)$ .

Using the property of radicals that states $√(a * b) = √a * √b$ , we can further simplify this as $√25 * √7$ . Since $√25 = 5$ , we get $5√7$ . This step is crucial in making the expression more manageable.

Step 2: Substitute the Simplified Radical Back into the Expression

Now that we have simplified $√175$ to $5√7$ , we can substitute this back into our original expression, $3√7 + 9√175$ . This gives us:

$3√7 + 9(5√7)$

Next, we perform the multiplication:

$3√7 + 45√7$

Now we have two terms with the same radical, $√7$ . This allows us to combine them, similar to combining like terms in algebra. This substitution process is vital for simplifying complex expressions.

Step 3: Combine Like Terms

We now have the expression $3√7 + 45√7$ . Since both terms have the same radical component ( $√7$ ), we can treat $√7$ as a common factor and combine the coefficients (the numbers in front of the radical). Think of it as combining “3 apples” and “45 apples.”

To combine these terms, we simply add the coefficients:

3 + 45 = 48

So, the simplified expression becomes:

$48√7$

This step highlights the importance of recognizing like terms in mathematical expressions. Combining like terms is a fundamental algebraic technique that simplifies expressions and makes them easier to work with.

Final Answer

Therefore, the simplified form of the expression $3√7 + 9√175$ is $48√7$ . This process involved simplifying radicals, substituting simplified forms back into the original expression, and combining like terms. Each step is crucial in arriving at the final, simplified answer. This final answer is now in its simplest form, with no further simplifications possible.

Practice Problems

To solidify your understanding of simplifying radical expressions, let’s look at a few practice problems. Working through these will give you hands-on experience and build your confidence in solving similar problems.

Problem 1: Simplify $2√12 + 5√27$

First, we need to simplify $√12$ and $√27$ individually.

For $√12$ , we look for the largest perfect square that divides 12. The largest perfect square is 4 (since 4 * 3 = 12). So, $√12 = √(4 * 3) = √4 * √3 = 2√3$ .

For $√27$ , the largest perfect square that divides 27 is 9 (since 9 * 3 = 27). Thus, $√27 = √(9 * 3) = √9 * √3 = 3√3$ .

Now, we substitute these simplified radicals back into the original expression:

$2√12 + 5√27 = 2(2√3) + 5(3√3) = 4√3 + 15√3$

Finally, we combine like terms:

$4√3 + 15√3 = 19√3$

So, the simplified expression is $19√3$ .

Problem 2: Simplify $4√50 - 3√8$

We start by simplifying $√50$ and $√8$ .

For $√50$ , the largest perfect square that divides 50 is 25 (since 25 * 2 = 50). So, $√50 = √(25 * 2) = √25 * √2 = 5√2$ .

For $√8$ , the largest perfect square that divides 8 is 4 (since 4 * 2 = 8). Thus, $√8 = √(4 * 2) = √4 * √2 = 2√2$ .

Substitute these back into the expression:

$4√50 - 3√8 = 4(5√2) - 3(2√2) = 20√2 - 6√2$

Combine like terms:

$20√2 - 6√2 = 14√2$

Therefore, the simplified expression is $14√2$ .

Problem 3: Simplify $√48 + 2√75 - √12$

Let's simplify each radical individually:

$√48 = √(16 * 3) = √16 * √3 = 4√3$
$√75 = √(25 * 3) = √25 * √3 = 5√3$
$√12 = √(4 * 3) = √4 * √3 = 2√3$

Substitute these back into the expression:

$√48 + 2√75 - √12 = 4√3 + 2(5√3) - 2√3 = 4√3 + 10√3 - 2√3$

Combine like terms:

$4√3 + 10√3 - 2√3 = 12√3$

Thus, the simplified expression is $12√3$ .

These practice problems should give you a clearer understanding of how to simplify radical expressions. The key is to identify the largest perfect square factors within the radicals and then combine like terms.

Tips and Tricks for Simplifying Radical Expressions

Simplifying radical expressions can become easier with practice and a few helpful tips and tricks. Here are some strategies to keep in mind as you work through these types of problems:

Always Look for Perfect Square Factors: The most critical step in simplifying radicals is to identify perfect square factors within the radicand. Start by checking common perfect squares like 4, 9, 16, 25, 36, 49, etc. This approach will help you break down the radical into simpler components.
Use Prime Factorization: If you are having trouble finding perfect square factors, try breaking down the radicand into its prime factors. For example, let’s consider $√180$ . The prime factorization of 180 is $2^2 * 3^2 * 5$ . From this, you can see that $√180 = √(2^2 * 3^2 * 5) = √(2^2) * √(3^2) * √5 = 2 * 3 * √5 = 6√5$ . This method is particularly useful for larger numbers.
Remember the Product Property of Radicals: The property $√(a * b) = √a * √b$ is your best friend when simplifying radicals. It allows you to separate a radical into multiple radicals, making it easier to identify and extract perfect square factors. For instance, $√72 = √(36 * 2) = √36 * √2 = 6√2$ .
Combine Like Terms Carefully: Just like in algebra, you can only combine terms that have the same radical. For example, $3√5 + 7√5$ can be combined because they both have $√5$ , but $3√5 + 7√2$ cannot be combined because they have different radicals. Make sure to pay attention to the radical part of the term when combining.
Simplify Radicals Before Combining: It’s generally best to simplify each radical individually before trying to combine terms. This makes the expression easier to manage and reduces the chances of making mistakes. For example, in the expression $2√8 + 3√32$ , simplify $√8$ to $2√2$ and $√32$ to $4√2$ before combining.
Practice Regularly: Like any mathematical skill, simplifying radicals becomes easier with practice. Work through a variety of problems to build your confidence and intuition. The more you practice, the faster and more accurately you’ll be able to simplify radicals.
Check Your Work: After simplifying an expression, take a moment to check your work. Ensure that you have correctly identified and extracted all perfect square factors and that you have combined like terms accurately. This simple step can help you catch and correct any errors.
Use Online Tools and Resources: There are many online tools and resources available that can help you practice and check your work. Websites like Khan Academy and Symbolab offer lessons, practice problems, and step-by-step solutions. These resources can be invaluable when you’re learning how to simplify radicals.

By following these tips and tricks, you’ll be well-equipped to simplify a wide range of radical expressions. Remember, the key is to break down the problem into manageable steps and to practice regularly.

Conclusion

In this article, we’ve covered the step-by-step process of simplifying the expression $3√7 + 9√175$ . We started by understanding the basics of radicals and then moved on to breaking down and simplifying $√175$ . We substituted the simplified form back into the original expression and combined like terms to arrive at the final answer, $48√7$ . Along the way, we also worked through practice problems and discussed valuable tips and tricks for simplifying radical expressions. With these tools and techniques, you should now feel confident in your ability to tackle similar problems.

For further learning and practice, you may find the resources at Khan Academy particularly helpful. They offer a wide range of math tutorials and exercises that can help you master simplifying radical expressions and other algebraic concepts.