Square Root of 278

The square root is the inverse of the square of a number. Since 278 is not a perfect square, its square root is expressed in both radical and exponential form. In the radical form, it is expressed as √278, whereas (278)^(1/2) in the exponential form. √278 ≈ 16.6733, 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 suitable for non-perfect square numbers, so 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 278 is broken down into its prime factors.

Step 1: Finding the prime factors of 278

Breaking it down, we get 2 x 139: 2^1 x 139^1

Step 2: Now we found out the prime factors of 278. Since 278 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 278 using prime factorization alone is not feasible.

The long division method is particularly used for non-perfect square numbers. 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 278, we need to group it as 78 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 x 1 = 1 is less than 2. Now the quotient is 1, and after subtracting 1 from 2, the remainder is 1.

Step 3: Bring down 78, the new dividend. Add the old divisor with the same number 1 + 1 to get 2, which will be our new divisor.

Step 4: Now, considering 2n as the new divisor, we need to find the value of n.

Step 5: We need 2n x n ≤ 178. Let us consider n as 6, then 26 x 6 = 156.

Step 6: Subtract 156 from 178; the difference is 22, 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 extend the dividend with two zeros. Now the new dividend is 2200.

Step 8: Find the new divisor, which is 333, because 333 x 6 = 1998.

Step 9: Subtracting 1998 from 2200, we get the result 202.

Step 10: Now the quotient is 16.6.

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

So the square root of √278 is approximately 16.67.

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 278 using the approximation method.

Step 1: We have to find the closest perfect squares around √278. The perfect square less than 278 is 256, and the perfect square greater than 278 is 289. √278 falls between 16 and 17.

Step 2: Apply the formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square) Using this formula: (278 - 256) / (289 - 256) ≈ 0.6875 Adding the decimal to the initial whole number: 16 + 0.6875 ≈ 16.6875, so the square root of 278 is approximately 16.688.

• Square root: A square root is the inverse of a square. Example: 4² = 16 and the inverse of the square is the square root; thus, √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, the positive square root, known as the principal square root, is often used in real-world applications.

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

• Decimal: A decimal is a number that contains a whole number and a fraction in a single number, such as 7.86, 8.65, and 9.42.