Square Root of 217

The square root is the inverse of the square of the number. 217 is not a perfect square. The square root of 217 is expressed in both radical and exponential form. In the radical form, it is expressed as √217, whereas (217)^(1/2) in the exponential form. √217 ≈ 14.73092, 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 217 is broken down into its prime factors:

Step 1: Finding the prime factors of 217 217 is a prime number, which means it only has two factors: 1 and 217.

Therefore, using prime factorization to simplify √217 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 217, we group it as 17 and 2.

Step 2: Now we need to find n whose square is less than or equal to 2. We can say n is ‘1’ because 1 × 1 is less than or equal to 2. The quotient is 1, and after subtracting, we have a remainder of 1.

Step 3: Bring down 17, making the new dividend 117. Add the old divisor with the same number, 1 + 1, to get 2, which will be our new divisor.

Step 4: We need to find the largest digit n such that 2n × n ≤ 117. Let n be 4, then 2 × 4 × 4 = 64.

Step 5: Subtract 64 from 117, the difference is 53, and the quotient is 14.

Step 6: Since the dividend is less than the divisor, we need to add a decimal point, allowing us to bring down pairs of zeros.

Step 7: Continue the long division process until you reach the desired decimal places, resulting in √217 ≈ 14.73.

The approximation method is another method for finding square roots. It is an easy way to find the square root of a given number. Now let us learn how to find the square root of 217 using the approximation method:

Step 1: Find the closest perfect square numbers to 217. The smallest perfect square less than 217 is 196 (14²), and the largest perfect square greater than 217 is 225 (15²). √217 falls between 14 and 15.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Largest perfect square - smallest perfect square). Using the formula (217 - 196) ÷ (225 - 196) = 21 ÷ 29 ≈ 0.724. Adding this to the smaller perfect square root, we get 14 + 0.724 ≈ 14.724.

• 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 number greater than 1 that has no positive divisors other than 1 and itself.

• Decimal approximation: The process of finding a decimal number that is close to an exact value.

• Long division method: A technique used to find the square root of non-perfect squares by dividing and averaging in successive steps.