Square Root of 4800

The square root is the inverse of the square of the number. 4800 is not a perfect square. The square root of 4800 is expressed in both radical and exponential form. In the radical form, it is expressed as √4800, whereas (4800)^(1/2) in the exponential form. √4800 ≈ 69.282, 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 4800 is broken down into its prime factors.

Step 1: Finding the prime factors of 4800 Breaking it down, we get 2 × 2 × 2 × 2 × 3 × 5 × 5 × 3: 2^4 × 3^2 × 5^2

Step 2: Now we found out the prime factors of 4800. The second step is to make pairs of those prime factors. Since 4800 is not a perfect square, the digits of the number can’t be completely grouped in pairs. Therefore, calculating √4800 using prime factorization directly 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 4800, we need to group it as 48 and 00.

Step 2: Now we need to find n whose square is less than or equal to 48. We can say n as ‘6’ because 6 × 6 = 36, which is less than 48. Now the quotient is 6 after subtracting 36 from 48, the remainder is 12.

Step 3: Now let us bring down the next pair of digits (00), making the new dividend 1200. Add the old divisor with the same number 6 + 6, we get 12 which will be our new divisor base.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 12n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 12n × n ≤ 1200. Let us consider n as 9, now 12 × 9 = 108, and 129 × 9 = 1161.

Step 6: Subtract 1161 from 1200, the difference is 39, and the quotient is 69.

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 3900.

Step 8: Now we need to find the new divisor, which is 692, because 692 × 5 = 3460.

Step 9: Subtracting 3460 from 3900, we get the result 440.

Step 10: Now the quotient is 69.5.

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

So the square root of √4800 is approximately 69.28.

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 4800 using the approximation method.

Step 1: Now we have to find the closest perfect square of √4800. The largest perfect square less than 4800 is 4761, and the smallest perfect square greater than 4800 is 4900. √4800 falls somewhere between 69 and 70.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (4800 - 4761) / (4900-4761) = 39/139 = 0.2806. Using the approximation, 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 69 + 0.2806 ≈ 69.28, so the square root of 4800 is approximately 69.28.

• 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.

• Perfect square: A perfect square is a number that can be expressed as the square of an integer. For example, 16 is a perfect square because it is 4^2.

• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example, 7.86, 8.65, and 9.42 are decimals.

• Prime factorization: The expression of a number as the product of its prime factors. For example, the prime factorization of 42 is 2 × 3 × 7.