Square Root of 11000

The square root is the inverse of the square of the number. 11000 is not a perfect square. The square root of 11000 is expressed in both radical and exponential form. In radical form, it is expressed as √11000, whereas (11000)^(1/2) in exponential form. √11000 ≈ 104.8809, 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 11000 is broken down into its prime factors.

Step 1: Finding the prime factors of 11000

Breaking it down, we get 2 x 2 x 2 x 5 x 5 x 11 x 5: 2^3 x 5^3 x 11

Step 2: Now we found out the prime factors of 11000. The next step is to make pairs of those prime factors. Since 11000 is not a perfect square, the digits of the number can’t be grouped into pairs completely. Therefore, calculating √11000 using prime factorization is not possible entirely.

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 11000, we need to group it as 00 and 110.

Step 2: Now we need to find n whose square is less than or equal to 110. We can say n as ‘10’ because 10^2 is less than or equal to 110. Now the quotient is 10 after subtracting 100 from 110, the remainder is 10.

Step 3: Now let us bring down 00, which makes the new dividend 1000. Add the old divisor 10 to itself to get 20, which will be our new divisor.

Step 4: Find a digit n such that (20n) × n is less than or equal to 1000.

Step 5: The next step is finding n = 4, as 204 × 4 = 816.

Step 6: Subtract 816 from 1000, the difference is 184, and the quotient is 104.

Step 7: Since the remainder is less than the divisor and we desire more precision, we add a decimal point and bring down two zeros to make the new dividend 18400.

Step 8: Find a new divisor by doubling the current quotient (104) and append a digit to form 1040_.

Step 9: Continue this process until you achieve the desired precision.

So the square root of √11000 is approximately 104.88.

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 11000 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √11000. The closest perfect squares are 10000 and 12100. √11000 falls somewhere between 100 and 110.

Step 2: Now we need to apply the formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square) Going by the formula (11000 - 10000) ÷ (12100 - 10000) = 1000 ÷ 2100 ≈ 0.476.

Using the formula, we identified the decimal point of our square root. The next step is adding the integer part of the square root to the decimal number, which is 104 + 0.88 = 104.88, so the square root of 11000 is approximately 104.88.

• Square root: A square root is the inverse of a square. 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.
   
• Perfect square: A perfect square is an integer that is the square of an integer. For example, 144 is a perfect square because it is 12^2.
   
• Exponent: An exponent refers to the number of times a number is multiplied by itself. For example, in 2^3, 3 is the exponent.
   
• Long division method: A method used to find the square root of non-perfect squares by dividing and finding the closest perfect square systematically.