Square Root of 151

The square root is the inverse of the square of the number. 151 is not a perfect square. The square root of 151 is expressed in both radical and exponential form. In the radical form, it is expressed as √151, whereas 151^(1/2) in the exponential form. √151 ≈ 12.28821, 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 151 is broken down into its prime factors:

Step 1: Finding the prime factors of 151 151 is a prime number, so it cannot be broken down further into other factors.

Since 151 is not a perfect square, calculating 151 using the prime factorization method is not applicable.

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 151, we need to group it as 51 and 1.

Step 2: Now we need to find a number whose square is less than or equal to 1. We can say this number is '1' because 1 × 1 is 1. The quotient is 1, and after subtracting 1 from 1, the remainder is 0.

Step 3: Now let us bring down 51, which is the new dividend. Add the old divisor with the same number: 1 + 1, which gives us 2, the new divisor.

Step 4: We get 2n as the new divisor. We need to find a value of n such that 2n × n is less than or equal to 51. Let us consider n as 2, so 22 × 2 = 44.

Step 5: Subtract 44 from 51; the difference is 7, and the quotient is 12.

Step 6: 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 700.

Step 7: Now we need to find the new divisor. Let us consider 24 as the divisor. If 24 and n = 2, we get 24 × 2 = 48, which is less than 700.

Step 8: Continue doing these steps until we get two numbers after the decimal point.

So the square root of √151 ≈ 12.29.

The approximation method is another way to find 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 151 using the approximation method.

Step 1: Now we have to find the closest perfect squares around √151. The smallest perfect square less than 151 is 144, and the largest perfect square greater is 169. √151 falls somewhere between 12 and 13.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (151 - 144) / (169 - 144) ≈ 0.28. Adding the integer part, the square root of 151 ≈ 12 + 0.28 = 12.28.

• 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.
   
• Long division method: A method used to find the square root (or other roots) of a number, especially useful for non-perfect squares.
   
• Decimal: If a number has a whole number and a fraction in a single number, it is called a decimal. For example, 7.86, 8.65, and 9.42 are decimals.