Square Root of 1420

The square root is the inverse of the square of the number. 1420 is not a perfect square. The square root of 1420 is expressed in both radical and exponential form. In the radical form, it is expressed as √1420, whereas (1420)^(1/2) is in exponential form. √1420 ≈ 37.6692, which is an irrational number because it cannot be expressed as 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 1420 is broken down into its prime factors.

Step 1: Finding the prime factors of 1420 Breaking it down, we get 2 x 2 x 5 x 71: 2² x 5 x 71

Step 2: Now we have found the prime factors of 1420. The second step is to make pairs of those prime factors. Since 1420 is not a perfect square, the digits of the number can’t be grouped in pairs completely.

Therefore, calculating the square root of 1420 using prime factorization does not give an exact 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 1420, we need to group it as 20 and 14.

Step 2: Now we need to find n whose square is less than or equal to 14. We can use n = 3 because 3² = 9, which is less than 14. The quotient is 3, and after subtracting 9 from 14, the remainder is 5.

Step 3: Now let us bring down 20, which is the new dividend. Add the old divisor with the same number, 3 + 3, to get 6, which will be our new divisor.

Step 4: The new divisor will be 6n. We need to find the value of n.

Step 5: Find 6n × n ≤ 520. Let us consider n as 8, now 68 x 8 = 544.

Step 6: Subtract 520 from 544, the difference is negative, so we try n as 7. 67 x 7 = 469.

Step 7: Subtracting 469 from 520 gives 51, and the quotient now is 37.

Step 8: 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 5100.

Step 9: Now we need to find the new divisor, which is 754, because 754 x 7 = 5278.

Step 10: Subtracting 5278 from 5100 gives a negative result, so we try n as 6.

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

So the square root of √1420 is approximately 37.67.

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 1420 using the approximation method.

Step 1: Now we have to find the closest perfect square to √1420.

The smallest perfect square below 1420 is 1369, and the largest perfect square above 1420 is 1444. √1420 falls somewhere between 37 and 38.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula: (1420 - 1369) / (1444 - 1369) = 51 / 75 = 0.68 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 37 + 0.68 = 37.68.

So the square root of 1420 is approximately 37.68.

• 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 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 always 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 process of expressing a number as a product of its prime factors. For example, the prime factorization of 1420 is 2² × 5 × 71.

• Long division method: A step-by-step process used to find the square root of a number, especially useful for numbers that are not perfect squares.