Square Root of 202

The square root is the inverse of squaring a number. 202 is not a perfect square. The square root of 202 is expressed in both radical and exponential form. In radical form, it is expressed as √202, whereas in exponential form it is (202)^(1/2). √202 ≈ 14.21267, which is an irrational number because it cannot be expressed as a fraction 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 like 202, 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 prime factorization of a number is the product of its prime factors. Now let us break down 202 into its prime factors:

Step 1: Finding the prime factors of 202 Breaking it down, we get 2 x 101: 2^1 x 101^1

Step 2: Since 202 is not a perfect square, the digits cannot be paired up evenly. Therefore, calculating 202 using prime factorization directly is not feasible for finding an exact square root.

The long division method is particularly useful for non-perfect square numbers. Here’s how to find the square root using the long division method, step by step:

Step 1: Group the digits of 202 from right to left. We have 02 and 2.

Step 2: Find n such that n^2 is less than or equal to 2. We choose n = 1 because 1^2 is 1, which is less than 2. The quotient is 1; after subtracting 1 from 2, the remainder is 1.

Step 3: Bring down 02, making the new dividend 102. Add the old divisor (1) to itself (1 + 1 = 2) to get the new divisor.

Step 4: The new divisor is 2n. Now find n such that 2n x n is less than or equal to 102. Consider n = 4; then 2 x 4 x 4 = 32.

Step 5: Subtract 32 from 102; the remainder is 70. The quotient is 14.

Step 6: Since the dividend is less than the divisor, add a decimal point and two zeros to the dividend, making it 7000.

Step 7: Find a new divisor. Use n = 2, so 286 x 2 = 572.

Step 8: Subtract 572 from 7000 to get 1428.

Step 9: Continue these steps to get more digits after the decimal point.

The square root of √202 ≈ 14.21.

The approximation method is an easy way to estimate the square root of a given number. Here’s how to find the square root of 202 using approximation:

Step 1: Identify the closest perfect squares around 202. The smallest perfect square less than 202 is 196, and the largest perfect square greater than 202 is 225. √202 falls between 14 and 15.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Using the formula: (202 - 196) / (225 - 196) = 6 / 29 ≈ 0.207 Add this decimal to the lower bound: 14 + 0.207 ≈ 14.207

Thus, the square root of 202 is approximately 14.21.

• Square root: A square root is the inverse operation of squaring a number. Example: 4^2 = 16, so the square root of 16 is √16 = 4.

• Irrational number: An irrational number is a number that cannot be expressed as a simple fraction p/q, where p and q are integers and q ≠ 0.

• Perfect square: A perfect square is a number that is the square of an integer. Example: 16 is a perfect square because it is 4^2.

• Prime factorization: The process of expressing a number as the product of its prime factors. Example: The prime factorization of 18 is 2 x 3^2.

• Decimal approximation: The representation of an irrational number rounded to a certain number of decimal places. Example: √2 ≈ 1.414.