Square Root of 2305

The square root is the inverse of the square of the number. 2305 is not a perfect square. The square root of 2305 is expressed in both radical and exponential form. In the radical form, it is expressed as √2305, whereas (2305)^(1/2) in the exponential form. √2305 ≈ 48.0104, 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 and approximation methods 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 2305 is broken down into its prime factors.

Step 1: Finding the prime factors of 2305 Breaking it down, we get 5 x 461: 5^1 x 461^1

Step 2: Now we found out the prime factors of 2305. The second step is to make pairs of those prime factors. Since 2305 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 2305 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 2305, we need to group it as 05 and 23.

Step 2: Now we need to find n whose square is less than or equal to 23. We can say n as ‘4’ because 4 x 4 = 16, which is lesser than 23. Now the quotient is 4, and after subtracting 16 from 23, the remainder is 7.

Step 3: Now let us bring down 05, which is the new dividend. Add the old divisor with the same number 4 + 4, we get 8, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 8n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 8n × n ≤ 705. Let us consider n as 8, now 8 x 8 x 8 = 704.

Step 6: Subtract 705 from 704, the difference is 1, and the quotient is 48.

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 100.

Step 8: Now we need to find the new divisor, which is 960 because 960 x 0 = 0

Step 9: Subtracting 0 from 100, we get the result 100.

Step 10: Now the quotient is 48.0

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal value, continue till the remainder is zero.

So the square root of √2305 is approximately 48.01.

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 2305 using the approximation method.

Step 1: Now we have to find the closest perfect square of √2305. The smallest perfect square less than 2305 is 2025, and the largest perfect square greater than 2305 is 2401. √2305 falls somewhere between 45 and 49.

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 (2305 - 2025) ÷ (2401 - 2025) = 0.76 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 45 + 0.76 = 45.76

So the square root of 2305 is approximately 48.01.

• Square root: A square root is the inverse of squaring a number. Example: 4^2 = 16, and the inverse of squaring 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 the positive square root that has more prominence due to its uses in the real world. That is why it is also known as the principal square root.

• Prime factorization: The expression of a number as the product of prime numbers. For example, the prime factorization of 2305 is 5 x 461.

• Long division method: A step-by-step method for finding the square root of a non-perfect square number by using division and subtraction.