Square Root of 2113

The square root is the inverse of the square of a number. 2113 is not a perfect square. The square root of 2113 is expressed in both radical and exponential forms. In the radical form, it is expressed as √2113, whereas in exponential form it is expressed as (2113)^(1/2). √2113 ≈ 45.973, which is an irrational number because it cannot be expressed as a fraction 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, methods such as long division and approximation are used. Let us now explore these 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 2113 can be broken down into its prime factors.

Step 1: Finding the prime factors of 2113 2113 is a prime number itself, which means it cannot be broken down into smaller prime factors. Since 2113 is not a perfect square, calculation using prime factorization is impossible.

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 2113, we need to group it as 13 and 21.

Step 2: Now we need to find n whose square is ≤ 21. We can say n is ‘4’ because 4 × 4 = 16, which is less than 21. Now the quotient is 4 after subtracting 21 - 16, the remainder is 5.

Step 3: Now let us bring down 13, which is the new dividend. Add the old divisor with the same number 4 + 4 to get 8, which will be our new divisor.

Step 4: The new divisor will be 8n. Now we need to find the value of n where 8n × n ≤ 513.

Step 5: The next step is finding a suitable n such that 8n × n ≤ 513. We find n is 5 because 85 × 5 = 425.

Step 6: Subtract 425 from 513, the difference is 88, and the quotient is 45.

Step 7: Since the new 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 8800.

Step 8: Multiply the new quotient by 2 (45 × 2 = 90), add a placeholder for n, making it 900n, then find n where 900n × n ≤ 8800. We find n is 9 because 909 × 9 = 8181.

Step 9: Subtracting 8181 from 8800, we get the result 619.

Step 10: The quotient is now 45.9.

Step 11: Continue these steps until we get two numbers after the decimal point. Continue until the remainder is zero or until the desired precision is achieved.

So the square root of √2113 is approximately 45.97.

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 2113 using the approximation method.

Step 1: Find the closest perfect squares to √2113. The smallest perfect square less than 2113 is 2025, and the largest perfect square more than 2113 is 2209. √2113 falls somewhere between 45 and 47.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Largest perfect square - smallest perfect square) Using the formula (2113 - 2025) / (2209 - 2025) = 88 / 184 ≈ 0.478

Using the formula, we identified the decimal point of our square root. The next step is adding the integer part to the decimal number, which is 45 + 0.478 ≈ 45.97, so the square root of 2113 is approximately 45.97.

• Square root: A square root is the inverse of a square. 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 as a simple fraction, meaning it cannot be expressed as p/q, where p and q are integers, and q ≠ 0.
   
• Prime number: A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. Example: 2113.
   
• Long division method: A method used to find the square root of non-perfect squares by breaking down the process into a series of steps involving division and averaging.
   
• Approximation: A method of finding an approximate value for a mathematical expression, often used when calculating square roots of non-perfect squares.