Square Root of 2000

The square root is the inverse of the square of the number. 2000 is not a perfect square. The square root of 2000 is expressed in both radical and exponential form. In the radical form, it is expressed as √2000, whereas in the exponential form it is (2000)^(1/2). √2000 ≈ 44.72136, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.

The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers where the long-division method and approximation method are used. Let us now learn the following methods:

• Prime factorization method
• Long division method
• Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 2000 is broken down into its prime factors.

Step 1: Finding the prime factors of 2000

Breaking it down, we get 2 × 2 × 2 × 2 × 5 × 5 × 5: 2^4 × 5^3

Step 2: Now we found out the prime factors of 2000. The second step is to make pairs of those prime factors. Since 2000 is not a perfect square, the digits of the number can’t be grouped into pairs evenly. Therefore, calculating √2000 using prime factorization directly is not possible.

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root using the long division method, step by step.

Step 1: To begin with, we need to group the numbers from right to left. In the case of 2000, we need to group it as 00 and 20.

Step 2: Now we need to find n whose square is less than or equal to 20. We can say n is ‘4’ because 4 × 4 = 16, which is less than or equal to 20. Now the quotient is 4. After subtracting 16 from 20, the remainder is 4.

Step 3: Bring down 00, making the new dividend 400. Double the current quotient, 4, to get 8, which will be part of our new divisor.

Step 4: We need to find a digit x such that 8x × x ≤ 400. Let us consider x as 4, now 84 × 4 = 336.

Step 5: Subtract 336 from 400, and the remainder is 64. The quotient is now 44.

Step 6: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 6400.

Step 7: The new divisor will be 88 (from 84) plus the next digit x. We need to find x such that 88x × x ≤ 6400. Let us consider x as 7, now 887 × 7 = 6209.

Step 8: Subtract 6209 from 6400, and the remainder is 191.

Step 9: The quotient is now 44.7.

Step 10: Continue doing these steps until we get two numbers after the decimal point or until the remainder is zero.

So the square root of √2000 is approximately 44.72.

The approximation method is another method for finding square roots; it is an easy method to find the square root of a given number. Now let us learn how to find the square root of 2000 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √2000. The smallest perfect square less than 2000 is 1936 (44^2) and the largest perfect square more than 2000 is 2025 (45^2). √2000 falls somewhere between 44 and 45.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula (2000 - 1936) ÷ (2025 - 1936) = 64 ÷ 89 ≈ 0.7191. Adding this to 44, we get 44 + 0.7191 ≈ 44.72, so the square root of 2000 is approximately 44.72.

• Square root: A square root is the inverse of a square. For example, 4^2 = 16 and the inverse of the square is the square root, that is √16 = 4.
   
• Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.
   
• Principal square root: A number has both positive and negative square roots; however, it is always the positive square root that has more prominence due to its uses in the real world. That is why it is known as the principal square root.
   
• Prime factorization: Prime factorization is expressing a number as the product of its prime factors.
   
• Decimal: If a number has a whole number and a fraction in a single number, it is called a decimal. For example, 7.86, 8.65, and 9.42 are decimals.