Breaking Down 200 into Prime Factors

Oct 13, 2025 by talynt.org 64 views

Understanding the Problem

To find the simplest form of $\sqrt{200}$ , we need to break down the number 200 into its prime factors. This will help us identify perfect squares that can be extracted from the square root.

Breaking Down 200 into Prime Factors

Let's start by finding the prime factors of 200.

200 = 2 × 2 × 2 × 5 × 5

We can rewrite this as:

200 = 2^3 × 5^2

Identifying Perfect Squares

Now that we have the prime factors of 200, we can identify the perfect squares.

2^2 = 4 (a perfect square)
5^2 = 25 (a perfect square)

We can rewrite $\sqrt{200}$ as:

$\sqrt{200}$ = $\sqrt{2^3 × 5^2}$

Simplifying the Square Root

Using the perfect squares we identified earlier, we can simplify the square root:

$\sqrt{200}$ = $\sqrt{2^3 × 5^2}$

= $\sqrt{(2^2 × 2) × (5^2)}$

= $\sqrt{(4 × 2) × 25}$

= $\sqrt{4 × 50}$

= $\sqrt{4} × \sqrt{50}$

= 2 × $\sqrt{50}$

Further Simplifying the Square Root

We can further simplify $\sqrt{50}$ by breaking it down into its prime factors:

50 = 2 × 5 × 5

We can rewrite $\sqrt{50}$ as:

$\sqrt{50}$ = $\sqrt{2 × 5 × 5}$

= $\sqrt{2 × 5^2}$

= 5 × $\sqrt{2}$

Final Simplification

Now that we have simplified $\sqrt{50}$ , we can substitute it back into our original expression:

$\sqrt{200}$ = 2 × $\sqrt{50}$

= 2 × 5 × $\sqrt{2}$

= 10 × $\sqrt{2}$

Conclusion

Therefore, the simplest form of $\sqrt{200}$ is 10 $\sqrt{2}$ .

Comparison with Answer Choices

Let's compare our answer with the answer choices provided:

A. $2 \sqrt{10}$

B. $10 \sqrt{2}$

C. $100 \sqrt{2}$

D. $20 \sqrt{10}$

Our answer, 10 $\sqrt{2}$ , matches answer choice B.

Final Thoughts

In this problem, we used the prime factorization method to simplify the square root of 200. We identified perfect squares and used them to simplify the expression. This method can be applied to other problems involving square roots and prime factorization.

Understanding Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16.

Frequently Asked Questions

Q: What is the simplest form of $\sqrt{200}$ ?

A: The simplest form of $\sqrt{200}$ is 10 $\sqrt{2}$ .

Q: How do I simplify a square root?

A: To simplify a square root, you need to break down the number into its prime factors. Identify perfect squares and use them to simplify the expression.

Q: What is the prime factorization of 200?

A: The prime factorization of 200 is 2^3 × 5^2.

Q: How do I identify perfect squares?

A: Perfect squares are numbers that can be expressed as the product of an even power of a prime number. For example, 4 is a perfect square because it can be expressed as 2^2.

Q: Can you give an example of simplifying a square root using prime factorization?

A: Let's simplify $\sqrt{200}$ using prime factorization:

$\sqrt{200}$ = $\sqrt{2^3 × 5^2}$

= $\sqrt{(2^2 × 2) × (5^2)}$

= $\sqrt{(4 × 2) × 25}$

= $\sqrt{4 × 50}$

= $\sqrt{4} × \sqrt{50}$

= 2 × $\sqrt{50}$

= 2 × 5 × $\sqrt{2}$

= 10 × $\sqrt{2}$

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

A: A square root is a value that, when multiplied by itself, gives the original number. Prime factorization is the process of breaking down a number into its prime factors.

Q: Can you give an example of a problem that involves both square roots and prime factorization?

A: Let's simplify $\sqrt{200}$ using both square roots and prime factorization:

$\sqrt{200}$ = $\sqrt{2^3 × 5^2}$

= $\sqrt{(2^2 × 2) × (5^2)}$

= $\sqrt{(4 × 2) × 25}$

= $\sqrt{4 × 50}$

= $\sqrt{4} × \sqrt{50}$

= 2 × $\sqrt{50}$

= 2 × 5 × $\sqrt{2}$

= 10 × $\sqrt{2}$

Q: How do I know if a number is a perfect square?

A: A number is a perfect square if it can be expressed as the product of an even power of a prime number.

Q: Can you give an example of a perfect square?

A: 16 is a perfect square because it can be expressed as 4^2.

Q: What is the difference between a perfect square and a prime factorization?

A: A perfect square is a number that can be expressed as the product of an even power of a prime number. Prime factorization is the process of breaking down a number into its prime factors.

Conclusion

Simplifying square roots involves breaking down the number into its prime factors and identifying perfect squares. By using prime factorization, you can simplify square roots and express them in their simplest form.