Square Root of 133

The square root is the inverse of the square of the number. 133 is not a perfect square. The square root of 133 is expressed in both radical and exponential form. In radical form, it is expressed as √133, whereas in exponential form it is expressed as (133)^(1/2). √133 ≈ 11.532, 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 133 is broken down into its prime factors:

Step 1: Finding the prime factors of 133 Breaking it down, we get 7 × 19: 7¹ × 19¹

Step 2: Now we found out the prime factors of 133. Since 133 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.
 

Therefore, calculating 133 using prime factorization is not feasible for finding an exact 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, group the numbers from right to left. In the case of 133, group it as 33 and 1.

Step 2: Now find n whose square is closest to or less than 1. We can say n as ‘1’ because 1 × 1 is equal to 1. Now the quotient is 1 and after subtracting 1-1 the remainder is 0.

Step 3: Bring down 33, 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 2n. We need to find the value of n such that 2n × n ≤ 33. If n is 5, then 25 ≤ 33.

Step 5: Subtract 25 from 33, the difference is 8. The quotient is 15.

Step 6: Since the dividend is less than the divisor, add a decimal point to the quotient and bring down two zeroes. Now the new dividend is 800.

Step 7: The new divisor is 30 because 305 × 5 = 1525.

Step 8: Subtracting 1525 from 8000, we get 6475.

Step 9: The quotient is now 11.5.

Step 10: Continue these steps until the desired precision is reached.

The square root of √133 is approximately 11.532.

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 133 using the approximation method.

Step 1: Find the closest perfect squares to √133. The smallest perfect square less than 133 is 121, and the largest perfect square greater than 133 is 144. √133 falls somewhere between 11 and 12.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula: (133 - 121) / (144 - 121) = 12/23 ≈ 0.522 Add this decimal to the integer part: 11 + 0.522 = 11.522

So, the square root of 133 is approximately 11.532.

• 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, which 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, but the positive square root is often used in real-world applications. This is known as the principal square root.
  
   
• Prime factorization: Prime factorization is expressing a number as the product of its prime factors.
  
   
• Approximation: Approximation is a method used to find a value that is close to but not exactly equal to the actual value, often used for non-perfect squares.