Square Root of 276

The square root is the inverse of the square of the number. 276 is not a perfect square. The square root of 276 is expressed in both radical and exponential form. In the radical form, it is expressed as √276, whereas (276)^(1/2) in the exponential form. √276 ≈ 16.6132, 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 276 is broken down into its prime factors.

Step 1: Finding the prime factors of 276

Breaking it down, we get 2 × 2 × 3 × 23: 2² × 3¹ × 23¹

Step 2: Now we found out the prime factors of 276. The next step is to make pairs of those prime factors. Since 276 is not a perfect square, the digits of the number cannot be grouped in pairs. Therefore, calculating 276 using prime factorization is not straightforward.

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 276, we need to group it as 76 and 2.

Step 2: Now we need to find n whose square is 2. We can say n as ‘1’ because 1 × 1 is lesser than or equal to 2. Now the quotient is 1, and after subtracting 1² from 2, the remainder is 1.

Step 3: Now let us bring down 76, which is the new dividend. Add the old divisor with the same number, 1 + 1, we get 2, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor; we need to find the value of n.

Step 5: The next step is finding 2n × n ≤ 176. Let us consider n as 6; now, 26 × 6 = 156.

Step 6: Subtract 176 from 156, the difference is 20, 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 2000.

Step 8: Now we need to find the new divisor which is 132 because 266 × 7 = 1862.

Step 9: Subtracting 1862 from 2000, we get the result 138.

Step 10: Now the quotient is 16.7.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal value, continue till the remainder is zero.

So the square root of √276 is approximately 16.61.

The approximation method is another method for finding the 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 276 using the approximation method.

Step 1: Now we have to find the closest perfect square of √276. The smallest perfect square less than 276 is 256 and the largest perfect square greater than 276 is 289. √276 falls somewhere between 16 and 17.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (276 - 256) ÷ (289 - 256) = 0.61. 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 + 0.61 = 16.61, so the square root of 276 is approximately 16.61.

• 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.

• 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 uses in the real world. That is the reason it is also known as the principal square root.

• Prime factorization: The process of expressing a number as the product of its prime factors.

• Perimeter: The perimeter is the total length around a two-dimensional shape, such as a rectangle or square.