Square Root of 5800

The square root is the inverse of the square of the number. 5800 is not a perfect square. The square root of 5800 is expressed in both radical and exponential form. In the radical form, it is expressed as √5800, whereas (5800)^(1/2) in the exponential form. √5800 ≈ 76.1577, 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 5800 is broken down into its prime factors.

Step 1: Finding the prime factors of 5800 Breaking it down, we get 2 x 2 x 2 x 5 x 5 x 29: 2^3 x 5^2 x 29

Step 2: Now we found out the prime factors of 5800. The second step is to make pairs of those prime factors. Since 5800 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating 5800 using prime factorization is impossible.

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 5800, we need to group it as 58 and 00.

Step 2: Now we need to find n whose square is less than or equal to 58. We can say n as ‘7’ because 7 x 7 = 49, which is less than 58. Now the quotient is 7 and after subtracting 58-49 the remainder is 9.

Step 3: Now let us bring down 00 which is the new dividend. Add the old divisor with the same number 7 + 7 we get 14 which will be our new divisor.

Step 4: The new divisor will be 14n. Now we need to find n such that 14n x n ≤ 900. Let us consider n as 6, now 146 x 6 = 876.

Step 5: Subtract 900 from 876, the difference is 24, and the quotient is 76.

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 2400.

Step 7: Now we need to find the new divisor that is 152 because 1524 x 4 = 6096 is too large, so we try 1523 x 3 = 4569 is too large, so 1522 x 2 = 3044 is too large, so 1521 x 1 = 1521 which works.

Step 8: Subtracting 1521 from 2400, we get the result 879.

Step 9: Now the quotient is approximately 76.15.

Step 10: 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 √5800 is approximately 76.16.

The approximation method is another method for finding the square roots, and 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 5800 using the approximation method.

Step 1: Now we have to find the closest perfect square of √5800.

The smallest perfect square less than 5800 is 5625 (75^2) and the largest perfect square more than 5800 is 5929 (77^2). √5800 falls somewhere between 75 and 77.

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 (5800 - 5625) ÷ (5929 - 5625) = 175 ÷ 304 ≈ 0.575.

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 75 + 0.575 ≈ 75.575, so the square root of 5800 is approximately 75.575.

• Square root: A square root is the inverse of a square. For 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, 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 a principal square root.

• Factors: Factors of a number are the integers that can be multiplied to produce the number. For example, factors of 12 are 1, 2, 3, 4, 6, and 12.

• Approximation: The process of finding a value that is close enough to the right answer, usually with some thought of the degree of accuracy required, is called approximation.