Square Root of 177

The square root is the inverse of the square of the number. 177 is not a perfect square. The square root of 177 is expressed in both radical and exponential form. In the radical form, it is expressed as √177, whereas (177)^(1/2) in the exponential form. √177 ≈ 13.3041, 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
  
   
• pproximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 177 is broken down into its prime factors.

Step 1: Finding the prime factors of 177 Breaking it down, we get 3 x 59.

Step 2: Since 177 is not a perfect square, the digits of the number can't be grouped in pairs.

Therefore, calculating 177 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 177, we need to group it as 77 and 1.

Step 2: Now we need to find n whose square is 1. We can say n is ‘1’ because 1 x 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 77, 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 ≤ 77; let us consider n as 3, now 23 x 3 = 69.

Step 6: Subtract 77 from 69, the difference is 8, and the quotient is 13.

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 800.

Step 8: Now we need to find the new divisor that is 266, because 266 x 3 = 798.

Step 9: Subtracting 798 from 800, we get the result 2.

Step 10: Now the quotient is 13.3.

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

So the square root of √177 is approximately 13.30.

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 177 using the approximation method.

Step 1: Now we have to find the closest perfect square to √177. The smallest perfect square less than 177 is 169, and the largest perfect square greater than 177 is 196. √177 falls somewhere between 13 and 14.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Applying the formula (177 - 169) ÷ (196 - 169) = 0.296. Using the formula we identify the decimal point of our square root. The next step is adding the value we got initially to the decimal number which is 13 + 0.296 = 13.296.

So the square root of 177 is approximately 13.30.

• 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, q is not equal to zero and p and q are integers.
   
• Radical form: The expression of a number under a square root symbol is called its radical form. Example: √177 is in its radical form.
   
• Approximation: This is a method used to find the square root of non-perfect square numbers by estimating between two perfect squares.
   
• Non-perfect square: A number that cannot be expressed as the square of an integer, for example, 177.