Square Root of 271

The square root is the inverse of the square of the number. 271 is not a perfect square. The square root of 271 is expressed in both radical and exponential form. In the radical form, it is expressed as √271, whereas (271)^(1/2) in the exponential form. √271 ≈ 16.46208, 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
• Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 271 is broken down into its prime factors:

Step 1: Finding the prime factors of 271 271 is a prime number, so it cannot be broken down further into other prime factors.

Step 2: Since 271 is a prime number, it cannot be grouped into pairs. Therefore, calculating √271 using the prime factorization method 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 271, we need to group it as 71 and 2.

Step 2: Now, we need to find n whose square is less than or equal to 2. We can choose n as ‘1’ because 1 × 1 = 1, which is less than or equal to 2. Now the quotient is 1, and after subtracting 1 from 2, the remainder is 1.

Step 3: Now let us bring down 71, which is the new dividend. Add the old divisor with the same number (1 + 1 = 2) to get the new divisor.

Step 4: The new divisor will be of the form 2n, and we need to find the value of n.

Step 5: The next step is finding 2n × n ≤ 171. Let us consider n as 6, now 26 × 6 = 156.

Step 6: Subtract 156 from 171, the difference is 15, and the quotient is 16.

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 1500.

Step 8: Now we need to find a new divisor. Trying n as 4, 328 × 4 = 1312.

Step 9: Subtracting 1312 from 1500, we get 188.

Step 10: Now the quotient is 16.4

Step 11: Continue doing these steps until we get two numbers after the decimal point, or until the remainder is minimized to zero.

So the square root of √271 ≈ 16.46

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 271 using the approximation method.

Step 1: Now we have to find the closest perfect square of √271. The smallest perfect square less than 271 is 256, and the largest perfect square greater than 271 is 289. √271 falls somewhere between 16 and 17.

Step 2: Now, we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Going by the formula: (271 - 256) / (289 - 256) = 15 / 33 ≈ 0.4545 Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number which is 16 + 0.4545 ≈ 16.4545, so the square root of 271 is approximately 16.46.

• Square root: A square root is the inverse of a square. For 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.

• Principal square root: A number has both positive and negative square roots; however, it is always the positive square root that has more prominence due to its use in the real world. This is why it is known as the principal square root.

• Prime number: A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. For example, 2, 3, 5, and 271 are prime numbers.

• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example, 7.86, 8.65, and 9.42 are decimals.