Square Root of 1570

The square root is the inverse of the square of a number. 1570 is not a perfect square. The square root of 1570 is expressed in both radical and exponential form. In the radical form, it is expressed as √1570, whereas in exponential form, it is (1570)^(1/2). √1570 ≈ 39.624, 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, it is not used for non-perfect square numbers where the long division method and approximation method are used. Let us now learn 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 1570 is broken down into its prime factors.

Step 1: Finding the prime factors of 1570 Breaking it down, we get 2 x 5 x 157.

Step 2: Now we found out the prime factors of 1570. Since 1570 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 1570 using prime factorization is not feasible for finding the exact square root.

The long division method is particularly used for non-perfect square numbers. In this method, we approximate the square root by dividing the number in a systematic way. Let us now learn how to find the square root using the long division method, step by step.

Step 1: Start by pairing the digits from right to left. In the case of 1570, we pair it as (15)(70).

Step 2: Find the largest number whose square is less than or equal to 15. This number is 3 because 3 x 3 = 9. Now the quotient is 3, and the remainder is 15 - 9 = 6.

Step 3: Bring down 70, making the new dividend 670. Double the quotient (3), giving us 6, and use this as part of the new divisor.

Step 4: Determine the next digit of the quotient (n) such that 6n x n is less than or equal to 670. The suitable n is 9, as 69 x 9 = 621.

Step 5: Subtract 621 from 670, leaving a remainder of 49.

Step 6: Add a decimal to the quotient, making it 39, and bring down two zeros, making the new dividend 4900.

Step 7: Use 2 x 39 = 78 as part of the new divisor and find n such that 78n x n is less than or equal to 4900. The suitable n is 6, as 786 x 6 = 4716.

Step 8: Subtract 4716 from 4900, leaving a remainder of 184.

Step 9: The quotient is 39.6. Continue this process to get more decimal places if needed.

Thus, √1570 ≈ 39.624.

The approximation method is another method for finding square roots. It is an easy method to estimate the square root of a given number. Now let us learn how to find the square root of 1570 using the approximation method.

Step 1: Find the closest perfect squares around 1570. The smallest perfect square less than 1570 is 1521 (39^2) and the largest perfect square greater than 1570 is 1600 (40^2). So, √1570 is between 39 and 40. 

Step 2: Use the formula:

(Given number - smaller perfect square) / (larger perfect square - smaller perfect square).

Applying the formula: (1570 - 1521) / (1600 - 1521) = 49 / 79 ≈ 0.62.

So, the square root of 1570 is approximately 39 + 0.62 = 39.62.

• Square root: A square root is the inverse of a square. Example: 4^2 = 16, and the inverse of the square is the square root, so √16 = 4.
   
• Irrational number: An irrational number 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, but the positive square root is usually more prominent due to its real-world applications. This is known as the principal square root.
   
• Prime factorization: The process of expressing a number as the product of its prime factors. For example, the prime factorization of 1570 is 2 x 5 x 157.
   
• Decimal: A number with a whole number and a fraction in a single number format, such as 7.86, 8.65, and 9.42, is called a decimal.