Square Root of 272

The square root is the inverse of squaring a number. 272 is not a perfect square. The square root of 272 can be expressed in both radical and exponential forms. In radical form, it is expressed as √272, whereas in exponential form it is expressed as (272)^(1/2). √272 ≈ 16.49242, which is an irrational number because it cannot be expressed as a ratio of two integers.

The prime factorization method is typically used for perfect square numbers. However, for non-perfect square numbers like 272, the long division method and approximation method are commonly used. Let's explore the following methods:

• Prime factorization method 
• Long division method 
• Approximation method

The prime factorization of a number is the product of its prime factors. Let's break down 272 into its prime factors:

Step 1: Finding the prime factors of 272 Breaking it down, we get 2 × 2 × 2 × 2 × 17: 2^4 × 17

Step 2: Now that we have found the prime factors of 272, the next step is to make pairs of those prime factors. Since 272 is not a perfect square, the digits cannot be grouped into pairs that form a perfect square, and thus calculating the square root using prime factorization is not straightforward.

The long division method is particularly useful for non-perfect square numbers. In this method, we find the closest perfect square number to the given number. Let's learn how to find the square root using the long division method, step by step:

Step 1: Start by grouping the digits in pairs from right to left. For 272, we group it as 72 and 2.

Step 2: Find a number n whose square is less than or equal to 2. Here, n is 1 because 1 × 1 ≤ 2. The quotient is 1, and the remainder is 1 after subtracting 1 from 2.

Step 3: Bring down 72 to make the new dividend 172. Double the previous divisor and append a digit (x) to form a new divisor, which is 2x.

Step 4: Find a digit x such that 2x × x ≤ 172. Let x = 6, then 26 × 6 = 156.

Step 5: Subtract 156 from 172, giving a remainder of 16. The quotient is 16.

Step 6: Bring down two zeros to make the new dividend 1600.

Step 7: The new divisor is 320 (since 32 × 5 = 1600), and the next digit of the quotient is 5.

Step 8: Continue this process until you achieve the desired precision. The square root of √272 is approximately 16.49.

The approximation method is a straightforward way to find square roots. Here's how to find the square root of 272 using approximation:

Step 1: Identify the closest perfect squares to 272. The smallest perfect square less than 272 is 256, and the largest perfect square greater than 272 is 289. Thus, √272 is between 16 and 17.

Step 2: Use the formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square) Applying the formula: (272 - 256) / (289 - 256) = 16 / 33 ≈ 0.48

Step 3: Add this decimal to the lower bound: 16 + 0.48 = 16.48 Thus, the approximate square root of 272 is 16.48.

• Square Root: The square root of a number is a value that, when multiplied by itself, gives the original number. Example: The square root of 16 is 4 because 4 × 4 = 16.

• Irrational Number: An irrational number is a number that cannot be expressed as a simple fraction or ratio of two integers. Example: √2 is an irrational number.

• Principal Square Root: The principal square root is the non-negative square root of a number. For example, the principal square root of 9 is 3.

• Prime Factorization: Prime factorization is expressing a number as the product of its prime numbers. Example: The prime factorization of 28 is 2 × 2 × 7.

• Approximation Method: The approximation method helps estimate the value of square roots for numbers that are not perfect squares, providing a close decimal value.