Simplify $\sqrt{200}$: Find The Equivalent Expression

Understanding Square Roots and Simplification

When we talk about simplifying square roots, we're essentially trying to find a more manageable way to represent a number that's under the radical symbol ($\sqrt{}$). The goal is to pull out any perfect square factors from under the radical. Think of it like this: if you have a group of items, and some of them can be neatly packaged into pairs (perfect squares), you can take those pairs out of the 'group' and leave the leftovers inside. For $\sqrt{200}$, we want to see if there are any perfect squares that are factors of 200. Perfect squares are numbers like 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, and so on – numbers you get when you multiply an integer by itself. By finding these perfect square factors, we can rewrite the original square root in a simpler form, making it easier to work with in further calculations or comparisons. This process is fundamental in algebra and is often one of the first steps in tackling more complex mathematical problems involving radicals.

Breaking Down $\sqrt{200}$: Finding Perfect Square Factors

Let's dive into simplifying $\sqrt{200}$. Our mission is to find the largest perfect square that divides evenly into 200. To do this, we can think about the factors of 200. We can start listing them, or better yet, we can look for known perfect squares. Does 4 divide 200? Yes, $200 \div 4 = 50$. So, we can write $\sqrt{200}$ as $\sqrt{4 \times 50}$. Since $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$, we can separate this into $\sqrt{4} \times \sqrt{50}$. We know that $\sqrt{4}$ is 2, so we have $2 \times \sqrt{50}$. Now, we need to check if $\sqrt{50}$ can be simplified further. Does 50 have any perfect square factors? Yes, 25 is a perfect square, and it divides into 50 ($50 \div 25 = 2$). So, we can rewrite $\sqrt{50}$ as $\sqrt{25 \times 2}$, which is $\sqrt{25} \times \sqrt{2}$. Since $\sqrt{25}$ is 5, this becomes $5 \times \sqrt{2}$. Putting it all together, our original expression $2 \times \sqrt{50}$ becomes $2 \times (5 \times \sqrt{2})$. Multiplying the numbers outside the radical, we get $10 \times \sqrt{2}$, or simply $10\sqrt{2}$.

Alternatively, we could have looked for the largest perfect square factor right away. What's the largest perfect square that goes into 200? We know $10^2 = 100$, and $200 \div 100 = 2$. So, 100 is a perfect square factor of 200. This means we can write $\sqrt{200}$ as $\sqrt{100 \times 2}$. Using the property of square roots again, this becomes $\sqrt{100} \times \sqrt{2}$. Since $\sqrt{100}$ is 10, our simplified expression is $10 \times \sqrt{2}$, which is $10\sqrt{2}$. This second method is generally more efficient because it requires fewer steps.

Evaluating the Options

Now that we've simplified $\sqrt{200}$ to $10\sqrt{2}$, let's compare this result to the given options:

• A. $2 \sqrt{10}$: To check if this is equivalent, we can square the number outside the radical and multiply it by the number inside. So, $2\sqrt{10} = \sqrt{2^2 \times 10} = \sqrt{4 \times 10} = \sqrt{40}$. Since $\sqrt{40}$ is not equal to $\sqrt{200}$, option A is incorrect.

• B. $10 \sqrt{2}$: This is exactly the result we obtained through our simplification process. Let's double-check: $10\sqrt{2} = \sqrt{10^2 \times 2} = \sqrt{100 \times 2} = \sqrt{200}$. This matches our original expression, so option B is the correct answer.

• C. $10 \sqrt{20}$: Similar to option A, let's square the outer number and multiply by the inner number: $10\sqrt{20} = \sqrt{10^2 \times 20} = \sqrt{100 \times 20} = \sqrt{2000}$. This is significantly larger than $\sqrt{200}$, so option C is incorrect.

• D. $100 \sqrt{2}$: Squaring the outer number and multiplying by the inner number gives: $100\sqrt{2} = \sqrt{100^2 \times 2} = \sqrt{10000 \times 2} = \sqrt{20000}$. This is much larger than $\sqrt{200}$, so option D is also incorrect.

Conclusion: The Equivalent Expression for $\sqrt{200}$

After systematically simplifying $\sqrt{200}$ and evaluating each of the provided options, we've confirmed that the only expression equivalent to $\sqrt{200}$ is $10\sqrt{2}$. This process highlights the importance of understanding how to simplify radicals by identifying and extracting perfect square factors. It's a fundamental skill that not only makes expressions more manageable but also forms the basis for more advanced mathematical operations. Keep practicing these simplification techniques, and you'll find that working with square roots becomes much more intuitive.

For further exploration of radical simplification and other mathematics topics, you can visit ** Khan Academy**.