Square Root of 11700

The square root is the inverse of the square of the number. 11700 is not a perfect square. The square root of 11700 is expressed in both radical and exponential form. In the radical form, it is expressed as √11700, whereas (11700)^(1/2) in the exponential form. √11700 ≈ 108.1665, 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 long-division method and approximation method are used. Let us now learn the following methods:

1. Prime factorization method
2. Long division method
3. Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 11700 is broken down into its prime factors

Step 1: Finding the prime factors of 11700 Breaking it down, we get 2 x 2 x 3 x 3 x 5 x 5 x 13: 2² x 3² x 5² x 13

Step 2: Now we found out the prime factors of 11700. The next step is to make pairs of those prime factors. Since 11700 is not a perfect square, therefore the digits of the number can’t be grouped into complete pairs.

Therefore, calculating √11700 using prime factorization gives us an approximate result of 108.1665.

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 11700, we need to group it as 00 and 117.

Step 2: Now we need to find n whose square is less than or equal to 117. We can say n as ‘10’ because 10 x 10 = 100, which is lesser than or equal to 117. Now the quotient is 10 after subtracting 117 - 100, the remainder is 17.

Step 3: Now let us bring down 00, which is the new dividend. Add the old divisor with the same number 10 + 10 to get 20, which will be our new divisor.

Step 4: The new divisor will be 20n. We need to find the value of n such that 20n x n ≤ 1700. Let us consider n as 8, now 208 x 8 = 1664.

Step 5: Subtract 1700 from 1664. The difference is 36, and the quotient is 108.

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 3600.

Step 7: Now we need to find the new divisor, which is 1082 because 1082 x 2 = 2164.

Step 8: Subtracting 2164 from 3600 we get 1436.

Repeat this process to get more decimal places. So the square root of √11700 is approximately 108.1665.

The approximation method is another method for finding the 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 11700 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √11700. The smallest perfect square less than 11700 is 11664 (108²), and the largest perfect square greater than 11700 is 11881 (109²). √11700 falls somewhere between 108 and 109.

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

Going by the formula (11700 - 11664) ÷ (11881 - 11664) = 36 ÷ 217 ≈ 0.1665. Using the formula, we identified the decimal point of our square root.

The next step is adding the value we got initially to the decimal number: 108 + 0.1665 = 108.1665, so the square root of 11700 is approximately 108.1665.

• Square root: A square root is the inverse of squaring a number. For example, 4² = 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 the reason it is also known as a principal square root.

• Prime factorization: Prime factorization involves expressing a number as a product of its prime factors. For example, the prime factorization of 11700 is 2 x 2 x 3 x 3 x 5 x 5 x 13.

• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example, 7.86, 8.65, and 9.42 are decimals.