Square Root of 1552

The square root is the inverse of the square of the number. 1552 is not a perfect square. The square root of 1552 is expressed in both radical and exponential form. In the radical form, it is expressed as √1552, whereas (1552)^(1/2) in the exponential form. √1552 ≈ 39.401, 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 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 1552 is broken down into its prime factors.

Step 1: Finding the prime factors of 1552 Breaking it down, we get 2 x 2 x 2 x 2 x 97: 2^4 x 97^1

Step 2: Now we have found the prime factors of 1552. The second step is to make pairs of those prime factors. Since 1552 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 1552 using prime factorization is not straightforward for finding the square root.

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 1552, we group it as 52 and 15.

Step 2: Now we need to find n whose square is closest to 15. We can say n as ‘3’ because 3 x 3 = 9 is less than 15. Now the quotient is 3; after subtracting 9 from 15, the remainder is 6.

Step 3: Bring down 52, making the new dividend 652. Add the old divisor with the same number 3 + 3 to get 6, which will be part of our new divisor.

Step 4: The new divisor is 6n. We need to find n such that 6n x n ≤ 652. Let us consider n as 9; now 69 x 9 = 621.

Step 5: Subtract 621 from 652; the difference is 31, and the quotient is 39.

Step 6: Since the dividend is less than the divisor, we add a decimal point, allowing us to add two zeroes to the dividend. The new dividend is 3100.

Step 7: Find the new divisor, which is 798 because 798 x 3 = 2394.

Step 8: Subtracting 2394 from 3100, we get the result 706.

Step 9: The quotient is now 39.3.

Step 10: Continue these steps until we have enough decimal places for accuracy.

So, the square root of √1552 ≈ 39.4

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 1552 using the approximation method.

Step 1: Identify the closest perfect squares to √1552.

The smallest perfect square less than 1552 is 1521 (39^2) and the largest perfect square more than 1552 is 1600 (40^2).

√1552 falls somewhere between 39 and 40.

Step 2: Apply the formula:

(Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

(1552 - 1521) / (1600 - 1521) = 31 / 79 ≈ 0.392

Using the formula, we identified the decimal.

Adding this to our initial whole number gives us 39 + 0.392 = 39.392, so the square root of 1552 ≈ 39.4

• 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, 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 usually the positive square root that is used, known as the principal square root.
   
• Prime factorization: Breaking down a composite number into a product of its prime factors.
   
• 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.