Square Root of 389

The square root is the inverse of the square of a number. 389 is not a perfect square. The square root of 389 is expressed in both radical and exponential forms. In radical form, it is expressed as √389, whereas in exponential form it is (389)^(1/2). √389 ≈ 19.723, 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 389 is broken down into its prime factors.

Step 1: Finding the prime factors of 389. Breaking it down, we get 389 = 389^1 (itself a prime number).

Step 2: Since 389 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating √389 using prime factorization is not possible.

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 389, we need to group it as 89 and 3.

Step 2: Now we need to find n whose square is less than or equal to 3. We can take n as ‘1’ because 1 × 1 = 1 is less than or equal to 3. Now, the quotient is 1, and after subtracting 1 from 3, the remainder is 2.

Step 3: Bring down 89, making the new dividend 289. Add the old divisor with the same number, 1 + 1, to get 2, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor; we need to find the value of n.

Step 5: Find 2n × n ≤ 289. Consider n as 7, now 27 × 7 = 189.

Step 6: Subtract 189 from 289, the difference is 100, and the quotient is 17.

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 zeros to the dividend. Now the new dividend is 10000.

Step 8: Find the new divisor that is 346 because 346 × 6 = 2076.

Step 9: Subtracting 2076 from 10000 gives the result 7924.

Step 10: Now the quotient is 19.7.

Step 11: Continue doing these steps until we get two numbers after the decimal point or until the remainder is zero.

So the square root of √389 is approximately 19.72.

The approximation method is another method for finding square roots, and it is an easy method to find the square root of a given number. Let us learn how to find the square root of 389 using the approximation method.

Step 1: Find the closest perfect squares around √389.

The closest perfect squares to 389 are 361 (19²) and 400 (20²).

√389 falls somewhere between 19 and 20.

Step 2: Apply the formula

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

Using the formula (389 - 361) / (400 - 361) = 28 / 39 ≈ 0.72.

Add this to the smaller whole number: 19 + 0.72 = 19.72.

So the square root of 389 is approximately 19.72.

• Square root: A square root is a value that, when multiplied by itself, gives the original number. For example, √16 = 4.
   
• Irrational number: An irrational number cannot be expressed as a simple fraction; for example, √2, √3, and √389.
   
• Prime number: A prime number is a natural number greater than 1 that has no divisors other than 1 and itself. For example, 2, 3, 389.
   
• Long division method: A technique used to find more precise approximations of square roots for non-perfect squares.
   
• Decimal: A number that contains a whole number part and a fractional part separated by a decimal point, such as 7.86 or 19.723.