Square Root of 448

The square root is the inverse of the square of the number. 448 is not a perfect square. The square root of 448 is expressed in both radical and exponential form. In the radical form, it is expressed as √448, whereas (448)^(1/2) in the exponential form. √448 ≈ 21.166, 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 typically used for perfect square numbers. However, for non-perfect square numbers, methods like the long-division method and the 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 448 is broken down into its prime factors:

Step 1: Finding the prime factors of 448. Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 7: 2^5 x 7.

Step 2: Now we found the prime factors of 448. The second step is to make pairs of those prime factors. Since 448 is not a perfect square, the digits of the number can’t be grouped into complete pairs.

Therefore, calculating 448 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 448, we need to group it as 48 and 4.

Step 2: Now we need to find n whose square is 4. We can say n is ‘2’ because 2 x 2 is equal to 4. Now the quotient is 2 after subtracting 4 from 4, the remainder is 0.

Step 3: Now let us bring down 48, which is the new dividend. Add the old divisor with the same number, 2 + 2, to get 4, which will be our new divisor.

Step 4: The new divisor will be part of finding 4n ≤ 48. Let us consider n as 1, now 4 x 1 = 4.

Step 5: Subtract 48 from 4; the difference is 44, and the quotient is 21.

Step 6: 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 4400.

Step 7: Now we need to find a new divisor that is 42 because 421 ✖ 9 = 3789.

Step 8: Subtracting 3789 from 4400, we get the result 611.

Step 9: Now the quotient is 21.1.

Step 10: Continue these steps until we get a precise enough value.

So the square root of √448 ≈ 21.166.

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 448 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √448. The smallest perfect square less than 448 is 441, and the largest perfect square greater than 448 is 484. √448 falls somewhere between 21 and 22.

Step 2: Now we need to apply the formula (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Going by the formula (448 - 441) ÷ (484 - 441) = 0.16.

Using the formula, we identified the decimal value.

The next step is adding the value we got initially to the decimal number, which is 21 + 0.16 = 21.16, so the square root of 448 is approximately 21.16.

• Square root: A square root is the inverse of a square. Example: 4^2 = 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, the positive square root is used more often due to its applications in the real world. This is known as the principal square root.

• Prime factorization: The process of breaking down a number into its prime factors. For example, the prime factorization of 448 is 2^5 x 7.

• Approximation method: A technique used to estimate the square root of a non-perfect square by identifying its closest perfect squares.