How do you simplify the square root of 200
The Correct Answer and Explanation is:
Correct Answer:
200=102\sqrt{200} = 10\sqrt{2}200=102
Step-by-step Explanation:
To simplify the square root of 200, you need to look for perfect square factors of 200. A perfect square is a number like 4, 9, 16, 25, 36, and so on. These numbers have whole numbers as their square roots.
Start by factoring 200:200=2×100200 = 2 \times 100200=2×100
Now check if 100 is a perfect square. Yes, it is, since:100=10\sqrt{100} = 10100=10
So:200=100×2\sqrt{200} = \sqrt{100 \times 2}200=100×2
Using the property of square roots:a×b=a×b\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}a×b=a×b
Apply this property:200=100×2\sqrt{200} = \sqrt{100} \times \sqrt{2}200=100×2200=10×2\sqrt{200} = 10 \times \sqrt{2}200=10×2200=102\sqrt{200} = 10\sqrt{2}200=102
This is the simplified form because 2\sqrt{2}2 is already in simplest radical form. It cannot be simplified further because 2 is a prime number and not a perfect square.
Why This Works:
Simplifying square roots involves factoring out the largest possible perfect square. The goal is to rewrite the number under the square root as a product where one factor is a perfect square. This makes it possible to take that part out of the square root, leaving the result in its simplest radical form.
In this example, 100 was the perfect square part of 200, and 2 was left inside the radical. By recognizing this, we arrived at a simplified expression that is easier to work with and often required in algebra and geometry problems.
