Square Root of 157

The square root is the inverse of the square of the number. 157 is not a perfect square. The square root of 157 is expressed in both radical and exponential form. In the radical form, it is expressed as √157, whereas (157)^(1/2) in the exponential form. √157 = 12.52996, 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 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 157 is broken down into its prime factors

Step 1: Finding the prime factors of 157 157 is a prime number, which means it cannot be further divided into other prime factors. Thus, prime factorization of 157 is 157 itself.

Step 2: Since 157 is not a perfect square, calculating 157 using prime factorization directly is not feasible 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 157, we need to group it as 57 and 1.

Step 2: Now we need to find n whose square is 1. We can say n as ‘1’ because 1×1 is lesser 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 57, which is the new dividend. Add the old divisor with the same number 1 + 1, we 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 ≤ 57. Let us consider n as 2, now 2×2×2 = 48.

Step 6: Subtract 57 from 48; the difference is 9, and the quotient is 12.

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 900.

Step 8: Now we need to find the new divisor that is 24 because 244 × 4 = 976.

Step 9: Subtracting 976 from 900, we get the result -76.

Step 10: Now the quotient is 12.5.

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

So the square root of √157 is approximately 12.53

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 157 using the approximation method.

Step 1: Now we have to find the closest perfect square of √157. The smallest perfect square of 157 is 144, and the largest perfect square of 157 is 169. √157 falls somewhere between 12 and 13.

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 (157 - 144) / (169 - 144) = 0.52 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 12 + 0.52 = 12.52.

So the square root of 157 is approximately 12.53

• 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, 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.
   
• Long division method: A method used to find the square root of non-perfect squares by dividing the number into groups and performing division iteratively.
   
• Approximation method: A method used to estimate the square root by comparing it to the nearest perfect squares and interpolating for a more precise value.