Square Root of 163

The square root is the inverse of the square of the number. 163 is not a perfect square. The square root of 163 is expressed in both radical and exponential form. In the radical form, it is expressed as √163, whereas (163)^(1/2) in the exponential form. √163 ≈ 12.7671, 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, for non-perfect square numbers such as 163, we use methods like the long-division method and the approximation method. 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 163 is broken down into its prime factors:


Step 1: Finding the prime factors of 163 Since 163 is a prime number itself, it cannot be broken down further into other prime factors.

Step 2: Since 163 is not a perfect square and is already a prime number, calculating its square root using prime factorization is not applicable.

The long division method is particularly used for non-perfect square numbers. 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 163, we need to consider it as 16 and 3.

Step 2: Find n whose square is less than or equal to 16. We can say n is '4' because 4 x 4 = 16. Now the quotient is 4, and after subtracting 16 - 16, the remainder is 0.

Step 3: Bring down the next pair, which is 3, making the new dividend 3. Add the old divisor with itself, 4 + 4, to get 8 as the new divisor.

Step 4: Determine n such that 8n x n is less than or equal to 300 (since we consider two decimal places, the new dividend is 300 after adding zeroes).

Step 5: Suppose n = 3, then 83 x 3 = 249.

Step 6: Subtract 249 from 300 to get 51. The quotient now becomes 12.7 (approx).

Step 7: Continue these steps until you reach the desired number of decimal places. The square root of √163 ≈ 12.7671.

The approximation method is another way to find the square roots. It is an easy method to find the square root of a given number. Let's learn how to find the square root of 163 using the approximation method.

Step 1: Identify the closest perfect squares around √163. The closest perfect squares to 163 are 144 (12²) and 169 (13²). √163 falls between 12 and 13.

Step 2: Use linear approximation to find the decimal part: (163 - 144)/(169 - 144) = 19/25 = 0.76

Add this decimal to the lower perfect square root: 12 + 0.76 = 12.76

Thus, the square root of 163 is approximately 12.76.

• Square root: The operation of finding a number that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, since 4 x 4 = 16.
   
• Irrational number: A number that cannot be expressed as a simple fraction; it has a non-repeating, non-terminating decimal expansion. For example, the square root of 163 is irrational.
   
• Prime number: A natural number greater than 1 that has no positive divisors other than 1 and itself. For example, 163 is a prime number.
   
• Radical expression: An expression that includes a square root, cube root, etc. For example, √163 is a radical expression. 
   
• Approximation: The process of finding a value that is close enough to the right answer, usually with some understanding of the error involved. For example, √163 ≈ 12.7671 is an approximation.