Square Root of 248

The square root is the inverse of squaring a number. 248 is not a perfect square. The square root of 248 is expressed in both radical and exponential forms. In the radical form, it is expressed as √248, whereas (248)^(1/2) in the exponential form. √248 ≈ 15.748, 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, for non-perfect square numbers, methods like 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 248 is broken down into its prime factors:

Step 1: Finding the prime factors of 248 Breaking it down, we get 2 x 2 x 2 x 31: 2^3 x 31

Step 2: Now that we have found the prime factors of 248, the second step is to make pairs of those prime factors. Since 248 is not a perfect square, the digits of the number can’t be grouped into pairs.

Therefore, calculating 248 using prime factorization is not feasible.

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 248, we need to group it as 48 and 2.

Step 2: Now we need to find n whose square is 2. We can say n as ‘1’ because 1 x 1 is lesser 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 48, which is the new dividend. Add the old divisor with the quotient 1 + 1 to get 2, which will be our new divisor.

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

Step 5: The next step is finding 2n × n ≤ 148. Let us consider n as 7, now 27 x 7 = 189.

Step 6: Subtract 148 from 189; since 189 is greater, we choose n=5. Thus, 25 x 5 = 125.

Step 7: Subtract 148 from 125, the difference is 23, and the quotient is 15.

Step 8: 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 2300.

Step 9: Now we need to find the new divisor, which is 310 because 310 x 7 = 2170.

Step 10: Subtracting 2170 from 2300, we get the result 130.

Step 11: Now the quotient is 15.7. Step 12: Continue doing these steps until we get two numbers after the decimal point. If there are no decimal values, continue until the remainder is zero.

So the square root of √248 is approximately 15.75.

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 248 using the approximation method.

Step 1: Now we have to find the closest perfect square of √248. The smallest perfect square less than 248 is 225, and the largest perfect square more than 248 is 256. √248 falls somewhere between 15 and 16.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (248 - 225) ÷ (256 - 225) = 23 ÷ 31 ≈ 0.74. 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 15 + 0.74 = 15.74.

So the square root of 248 is approximately 15.74.

• Square root: A square root is the inverse of squaring a number. For example, 4^2 = 16, and the square root is the inverse, √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 is often used in practical applications, known as the principal square root.
   
• Perfect square: A perfect square is a number that is the square of an integer. For example, 16 is a perfect square because it is 4^2.
   
• Long division method: A method used to find the square root of non-perfect squares by dividing the number into groups and using a step-by-step division process to approximate the square root.