Square Root of 1458

The square root is the inverse of the square of the number. 1458 is not a perfect square. The square root of 1458 is expressed in both radical and exponential form. In the radical form, it is expressed as √1458, whereas (1458)^(1/2) in the exponential form. √1458 ≈ 38.16625, 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:

• 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 1458 is broken down into its prime factors.

Step 1: Finding the prime factors of 1458 Breaking it down, we get 2 x 3 x 3 x 3 x 3 x 3 x 3: 2^1 x 3^6

Step 2: Now we found out the prime factors of 1458. The second step is to make pairs of those prime factors. Since 1458 is not a perfect square, therefore the digits of the number can’t be grouped in complete pairs.

Therefore, calculating the square root of 1458 using prime factorization is possible but will not yield an integer result.

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 1458, we need to group it as 58 and 14.

Step 2: Now we need to find n whose square is less than or equal to 14. We can say n is ‘3’ because 3 x 3 = 9 is lesser than or equal to 14. Now, the quotient is 3, and after subtracting 9 from 14, the remainder is 5.

Step 3: Now let us bring down 58, which is the new dividend. Add the old divisor with the same number, 3 + 3 = 6, which will be our new divisor.

Step 4: We need to find a digit X such that (60 + X) × X is less than or equal to 558.

Step 5: Trying with X = 9, we get (60 + 9) × 9 = 549. Subtract 549 from 558; the remainder is 9.

Step 6: Since there is a remainder, we can add a decimal point to the quotient and bring down pairs of zeros to continue the division.

Step 7: Continue this method until you achieve the desired precision.

So, the square root of √1458 is approximately 38.166.

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 1458 using the approximation method.

Step 1: Find the closest perfect squares of √1458.

The smallest perfect square less than 1458 is 1444, and the largest perfect square greater than 1458 is 1521. √1458 falls somewhere between 38 and 39.

Step 2: Use the formula: (Given number - smallest perfect square) ÷ (greater perfect square - smallest perfect square)

Using the formula (1458 - 1444) ÷ (1521 - 1444) = 14 ÷ 77 ≈ 0.1818 Adding this to the integer part of the square root of the smaller perfect square: 38 + 0.1818 ≈ 38.18

Therefore, the square root of 1458 is approximately 38.18.

• Square root: A square root of a number is one of its two equal factors. For example, 6 is a square root of 36 because 6 × 6 = 36.

• 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.

• Approximation: A method of finding a number close to the exact value. For example, the square root of 1458 is approximately 38.166.

• Long division method: A technique for dividing large numbers by breaking them down into a series of easier steps. It is often used to calculate square roots.

• Prime factorization: The process of expressing a number as the product of its prime factors. For example, 1458 = 2 x 3^6.