square root of 200

The square root of 200 is a value that, when multiplied by itself, gives the number 200. The square root of 200 is denoted as √200.

Answer:

The square root of 200 is approximately 14.1421.

Method: Hints

To find the square root of 200, we can use the following steps:

1. Estimation: Start by estimating a number that, when squared, is close to 200.
2. Prime Factorization: Break down 200 into its prime factors.
3. Pairing Prime Factors: Group the prime factors into pairs.
4. Simplify Square Root: Use the pairs to simplify the square root.
5. Decimal Approximation: If necessary, use a calculator to find a decimal approximation.

List of Calculation Steps and Descriptions

1. Estimation:

   • Guess a number that might be the square root of 200. For example, we know that 14^2 = 196 and 15^2 = 225. So, the square root of 200 must be between 14 and 15.

2. Prime Factorization:

   • Write 200 as a product of its prime factors: $200 = 2 \times 2 \times 2 \times 5 \times 5$.

3. Pairing Prime Factors:

   • Group the prime factors into pairs: $(2 \times 2) \times (2) \times (5 \times 5)$.

4. Simplify Square Root:

   • Take one number from each pair out of the square root: $\sqrt{200} = \sqrt{2 \times 2 \times 2 \times 5 \times 5} = 2 \times 5 \times \sqrt{2} = 10\sqrt{2}$.

5. Decimal Approximation:

   • Use a calculator to find the decimal approximation of $\sqrt{2}$, which is approximately 1.41421. Multiply this by 10 to get the approximate square root of 200: $10 \times 1.41421 \approx 14.1421$.

Verification

To check if our answer is correct, we can square the approximate value of the square root:

$(14.1421)^2 \approx 200.00004161$

Since this is very close to 200, we can be confident that our approximation is correct.

Related Knowledge Points and Detailed Explanation

• Square Roots: The square root of a number is a value that, when multiplied by itself, gives the original number.
• Prime Factorization: Breaking down a composite number into a product of its prime factors.
• Pairs in Square Roots: When simplifying square roots, each pair of prime factors comes out of the square root as a single number.
• Rational vs. Irrational Numbers: The square root of a non-perfect square (like 200) is an irrational number, meaning it cannot be expressed as a simple fraction and its decimal representation is non-repeating and non-terminating.
• Approximation: For non-perfect squares, we often use decimal approximations to express the square root.

By understanding these concepts, we can find the square root of any positive number, whether it is a perfect square or not.