Square Root of 5700

The square root is the inverse of the square of the number. 5700 is not a perfect square. The square root of 5700 is expressed in both radical and exponential forms. In radical form, it is expressed as √5700, whereas in exponential form it is expressed as (5700)^(1/2). √5700 = 75.497, 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 5700 is broken down into its prime factors:

Step 1: Finding the prime factors of 5700 Breaking it down, we get 2 × 2 × 3 × 5 × 5 × 19: 2^2 × 3^1 × 5^2 × 19^1

Step 2: Now we found out the prime factors of 5700. The second step is to make pairs of those prime factors. Since 5700 is not a perfect square, the digits of the number can’t be grouped in pairs perfectly.

Therefore, calculating √5700 using prime factorization is not straightforward without approximation.

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 5700, we need to group it as 57 and 00.

Step 2: Now we need to find n whose square is less than or equal to 57. We can say n is ‘7’ because 7 × 7 = 49, which is less than 57. Now the quotient is 7, and after subtracting 49 from 57, the remainder is 8.

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

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 140 as the new partial divisor, and we need to find the value of n.

Step 5: The next step is finding 140n × n ≤ 800. Let us consider n as 5, and now 140 × 5 = 700.

Step 6: Subtract 800 from 700, the difference is 100, and the quotient is 75.

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 10000.

Step 8: Now we need to find the new divisor, which is 150, because 1500 × 6 = 9000.

Step 9: Subtracting 9000 from 10000 gives the result 1000.

Step 10: Now the quotient is 75.49.

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

So the square root of √5700 is approximately 75.497.

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 5700 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √5700.

The smallest perfect square less than 5700 is 5625, and the largest perfect square greater than 5700 is 5776. √5700 falls somewhere between 75 and 76.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Going by the formula: (5700 - 5625) ÷ (5776 - 5625) = 75 ÷ 151 ≈ 0.497.

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 + 0.497 = 75.497, so the square root of 5700 is approximately 75.497.

• Square root: A square root is the inverse of a square. Example: 4² = 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 the principal square root.

• Prime factorization: It is a method of expressing a number as a product of its prime factors. For example, the prime factorization of 18 is 2 × 3 × 3.

• Long division method: A technique used to find the square root of non-perfect squares by dividing the number into parts to find an approximate value.