Go Step By Step To Reduce The Radical: 200 \sqrt{200} 200 Break Down Into Its Prime Factors: 4 ⋅ 5 2 ⋅ 2 = 4 ⋅ 5 2 ⋅ 2 \sqrt{4 \cdot 5^2 \cdot 2} = \sqrt{4} \cdot \sqrt{5^2} \cdot \sqrt{2} 4 ⋅ 5 2 ⋅ 2 = 4 ⋅ 5 2 ⋅ 2 Simplify: 2 ⋅ 5 ⋅ 2 = 10 2 2 \cdot 5 \cdot \sqrt{2} = 10\sqrt{2} 2 ⋅ 5 ⋅ 2 = 10 2 Answer:

Mar 12, 2025 by ADMIN 319 views

$**Simplifying Radicals: A Step-by-Step Guide to Reducing $\sqrt{200}$**$

Introduction

Radicals, also known as square roots, are an essential concept in mathematics. They are used to represent the value of a number that, when multiplied by itself, gives the original number. In this article, we will explore how to simplify radicals, specifically the radical $\sqrt{200}$ . We will break down the process into manageable steps, making it easier to understand and apply.

Breaking Down the Radical

The first step in simplifying the radical $\sqrt{200}$ is to break it down into its prime factors. This involves expressing the number under the radical sign as a product of its prime factors. In this case, we can write $\sqrt{200}$ as $\sqrt{4 \cdot 5^2 \cdot 2}$ .

\sqrt{200} = \sqrt{4 \cdot 5^2 \cdot 2}

Simplifying the Radical

Now that we have broken down the radical into its prime factors, we can simplify it by taking the square root of each factor. We can rewrite the radical as $\sqrt{4} \cdot \sqrt{5^2} \cdot \sqrt{2}$ .

\sqrt{4 \cdot 5^2 \cdot 2} = \sqrt{4} \cdot \sqrt{5^2} \cdot \sqrt{2}

Simplifying Each Factor

Now that we have the radical broken down into its prime factors, we can simplify each factor by taking the square root. The square root of 4 is 2, the square root of $5^2$ is 5, and the square root of 2 is $\sqrt{2}$ .

\sqrt{4} = 2
\sqrt{5^2} = 5
\sqrt{2} = \sqrt{2}

Combining the Simplified Factors

Now that we have simplified each factor, we can combine them to get the final simplified radical. We multiply the simplified factors together to get $2 \cdot 5 \cdot \sqrt{2}$ .

2 \cdot 5 \cdot \sqrt{2} = 10\sqrt{2}

Conclusion

In this article, we have explored how to simplify radicals, specifically the radical $\sqrt{200}$ . We broke down the radical into its prime factors, simplified each factor, and combined the simplified factors to get the final simplified radical. By following these steps, we can simplify radicals and make them easier to work with.

Tips and Tricks

When breaking down a radical into its prime factors, make sure to express the number under the radical sign as a product of its prime factors.
When simplifying each factor, make sure to take the square root of each factor.
When combining the simplified factors, make sure to multiply the simplified factors together.

Common Mistakes

Not breaking down the radical into its prime factors before simplifying.
Not taking the square root of each factor when simplifying.
Not multiplying the simplified factors together when combining them.

Real-World Applications

Simplifying radicals has many real-world applications. For example, in physics, radicals are used to represent the value of a quantity that, when multiplied by itself, gives the original quantity. In engineering, radicals are used to represent the value of a quantity that, when multiplied by itself, gives the original quantity. In finance, radicals are used to represent the value of a quantity that, when multiplied by itself, gives the original quantity.

Conclusion

Q: What is a radical?

A: A radical, also known as a square root, is a mathematical operation that represents the value of a number that, when multiplied by itself, gives the original number.

Q: Why do we need to simplify radicals?

A: Simplifying radicals is necessary because it makes them easier to work with and understand. By simplifying radicals, we can make complex calculations and expressions more manageable.

Q: How do I simplify a radical?

A: To simplify a radical, you need to break it down into its prime factors, simplify each factor, and combine the simplified factors.

Q: What are prime factors?

A: Prime factors are the prime numbers that multiply together to give the original number. For example, the prime factors of 12 are 2 and 3, because 2 x 2 x 3 = 12.

Q: How do I find the prime factors of a number?

A: To find the prime factors of a number, you need to divide the number by the smallest prime number (2) and continue dividing by prime numbers until you reach 1.

Q: What is the difference between a radical and a square root?

A: A radical and a square root are the same thing. The term "radical" is often used to refer to the symbol √, while the term "square root" is used to refer to the value that the radical represents.

Q: Can I simplify a radical with a variable?

A: Yes, you can simplify a radical with a variable. However, you need to follow the same steps as simplifying a radical with a number.

Q: How do I simplify a radical with a variable and a number?

A: To simplify a radical with a variable and a number, you need to break down the radical into its prime factors, simplify each factor, and combine the simplified factors.

Q: What are some common mistakes to avoid when simplifying radicals?

A: Some common mistakes to avoid when simplifying radicals include:

Not breaking down the radical into its prime factors before simplifying.
Not taking the square root of each factor when simplifying.
Not multiplying the simplified factors together when combining them.

Q: How do I know if a radical can be simplified?

A: A radical can be simplified if it can be broken down into its prime factors and simplified. If the radical cannot be broken down into its prime factors, it cannot be simplified.

Q: Can I simplify a radical with a negative number?

A: Yes, you can simplify a radical with a negative number. However, you need to follow the same steps as simplifying a radical with a positive number.

Q: How do I simplify a radical with a negative number and a variable?

A: To simplify a radical with a negative number and a variable, you need to break down the radical into its prime factors, simplify each factor, and combine the simplified factors.

Conclusion

In conclusion, simplifying radicals is an essential concept in mathematics. By understanding the basics of simplifying radicals, you can make complex calculations and expressions more manageable. Remember to break down the radical into its prime factors, simplify each factor, and combine the simplified factors to simplify a radical.