Square Root of 139

The square root is the inverse of the square of the number. 139 is not a perfect square. The square root of 139 is expressed in both radical and exponential form. In the radical form, it is expressed as √139, whereas (139)^(1/2) in the exponential form. √139 ≈ 11.79, 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 139 is broken down into its prime factors.

Step 1: Finding the prime factors of 139 139 is a prime number itself, so its only prime factor is 139.

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Step 2: Since 139 is not a perfect square, calculating √139 using prime factorization requires taking the square root of the prime number, which remains √139.

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 139, we need to group it as 39 and 1.

Step 2: Now we need to find n whose square is 1. We can say n is '1' because 1 × 1 is less than or equal to 1. Now the quotient is 1 after subtracting 1-1, the remainder is 0.

Step 3: Now let us bring down 39, which is the new dividend. 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: The next step is finding 2n × n ≤ 39. Let us consider n as 1, now 2 × 1 × 1 = 2

Step 6: Subtracting 39 from 2 gives the difference of 37, and the quotient is 11.

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 zeroes to the dividend. Now the new dividend is 3700.

Step 8: Now we need to find the new divisor, which is 119 because 219 × 9 = 1971.

Step 9: Subtracting 1971 from 3700, we get the result 1729.

Step 10: Now the quotient is 11.9

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

So the square root of √139 is approximately 11.79.

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 139 using the approximation method.

Step 1: Now we have to find the closest perfect square of √139. The smallest perfect square less than 139 is 121 and the largest perfect square greater than 139 is 144. √139 falls somewhere between 11 and 12.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (139 - 121) / (144 - 121) = 0.783 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 11 + 0.783 = 11.783.

So the square root of 139 is approximately 11.783.

Students do make mistakes while finding the square root, such as forgetting about the negative square root, skipping long division methods, etc. Now let us look at a few of those mistakes that students tend to make in detail.

• 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.
   
• Prime number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
   
• 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.
   
• Long division method: The long division method is a way to find the square root of a non-perfect square number by dividing it into pairs of digits starting from the right.