Square Root of 11016

The square root is the inverse of the square of a number. 11016 is not a perfect square. The square root of 11016 is expressed in both radical and exponential form. In the radical form, it is expressed as √11016, whereas (11016)^(1/2) in the exponential form. √11016 ≈ 104.941, 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, for non-perfect square numbers, the prime factorization method is not used; instead, 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 11016 is broken down into its prime factors.

Step 1: Finding the prime factors of 11016

Breaking it down, we get 2 × 2 × 2 × 3 × 3 × 3 × 3 × 17: 2^3 × 3^4 × 17

Step 2: Now we found out the prime factors of 11016. The second step is to make pairs of those prime factors. Since 11016 is not a perfect square, the digits of the number can’t be grouped in pairs completely. Therefore, calculating 11016 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 11016, we need to group it as 16 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 = 100 is lesser than or equal to 110. Now the quotient is 10 after subtracting 110 - 100, the remainder is 10.

Step 3: Now let us bring down 16, which is the new dividend. Add the old divisor with the same number: 10 + 10 = 20, which will be our new divisor.

Step 4: The new divisor will be 20n. We need to find the value of n that satisfies 20n × n ≤ 1016. Let us consider n as 5; now 205 × 5 = 1025.

Step 5: Subtract 1016 from 1025, the difference is -9. As it went negative, we take n as 4, now 204 × 4 = 816.

Step 6: Subtract 1016 from 816, the difference is 200, and the quotient becomes 104.

Step 7: 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 20000.

Step 8: Now we need to find the new divisor, which is 1049, because 1049 × 9 = 9441.

Step 9: Subtracting 9441 from 20000, we get the result 10559.

Step 10: Continue repeating these steps until we get two numbers after the decimal point or a suitable approximation.

So the square root of √11016 is approximately 104.941.

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 11016 using the approximation method.

Step 1: Now we have to find the closest perfect squares around √11016. The smallest perfect square less than 11016 is 10404 and the largest perfect square greater than 11016 is 11236. √11016 falls somewhere between 102 and 106.

Step 2: Now we need to apply the interpolation formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Applying this formula, (11016 - 10404) ÷ (11236 - 10404) = 0.612.

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 which is 104 + 0.612 = 104.612, so the square root of 11016 is approximately 104.612.

• 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, which 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 a number that can be expressed as the product of an integer with itself. For example, 36 is a perfect square because 6 × 6 = 36.
   
• Long Division Method: A technique used to find the square root of non-perfect squares by dividing the number into groups and finding the square root step by step.
   
• Approximation Method: A method to estimate the square root of a number by finding the closest perfect squares and using interpolation to find an approximate value.